Chapter 4 Fractions & Decimals

Given:

The fractions: $\frac{11}{9}$, $\frac{11}{7}$, $\frac{11}{10}$, $\frac{11}{6}$.

To Find:

The smallest fraction among the given options.

Solution:

All the given fractions have the same numerator, which is 11.

When comparing fractions with the same numerator, the fraction with the larger denominator is smaller, and the fraction with the smaller denominator is larger.

The denominators of the given fractions are 9, 7, 10, and 6.

Let's compare these denominators to find the largest one:

$6 < 7 < 9 < 10$

The largest denominator is 10.

The fraction with the largest denominator (10) is $\frac{11}{10}$.

Therefore, the smallest fraction among the given options is $\frac{11}{10}$.

The correct option is (C).

Given:

The decimal number: 0.7625

The intervals given as options:

(A) 0.7 and 0.76

(B) 0.77 and 0.78

(C) 0.76 and 0.761

(D) 0.76 and 0.763

To Find:

Which interval the number 0.7625 lies between.

Solution:

To determine which interval 0.7625 lies between, we compare it with the endpoints of each interval.

Let's evaluate each option:

(A) 0.7 and 0.76

The interval is $(0.7, 0.76)$.

Is $0.7 < 0.7625 < 0.76$? We can write 0.7 as 0.7000 and 0.76 as 0.7600.

So, is $0.7000 < 0.7625 < 0.7600$? No, 0.7625 is greater than 0.7600.

(B) 0.77 and 0.78

The interval is $(0.77, 0.78)$.

Is $0.77 < 0.7625 < 0.78$? We can write 0.77 as 0.7700.

So, is $0.7700 < 0.7625$? No, 0.7625 is less than 0.7700.

(C) 0.76 and 0.761

The interval is $(0.76, 0.761)$.

Is $0.76 < 0.7625 < 0.761$? We can write 0.76 as 0.7600 and 0.761 as 0.7610.

So, is $0.7600 < 0.7625 < 0.7610$? No, 0.7625 is greater than 0.7610.

(D) 0.76 and 0.763

The interval is $(0.76, 0.763)$.

Is $0.76 < 0.7625 < 0.763$? We can write 0.76 as 0.7600 and 0.763 as 0.7630.

So, is $0.7600 < 0.7625 < 0.7630$?

Comparing 0.7600 and 0.7625: The digits are the same up to the hundredths place. In the thousandths place, 2 is greater than 0. So, $0.7625 > 0.7600$.

Comparing 0.7625 and 0.7630: The digits are the same up to the hundredths place. In the thousandths place, 2 is less than 3. So, $0.7625 < 0.7630$.

Thus, $0.7600 < 0.7625 < 0.7630$, which means $0.76 < 0.7625 < 0.763$.

The number 0.7625 lies between 0.76 and 0.763.

The correct option is (D).

Given:

The decimal number 8.125.

To Find:

The fraction equal to the decimal 8.125.

Solution:

To convert a decimal to a fraction, we can write the decimal number over a power of 10.

The number of digits after the decimal point determines the power of 10 in the denominator.

In the decimal 8.125, there are 3 digits after the decimal point (1, 2, and 5).

So, the denominator will be $10^3 = 1000$.

The number 8.125 can be written as the fraction $\frac{8125}{1000}$.

Now, we need to simplify this fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Both 8125 and 1000 are divisible by 5 because they end in 5 and 0.

$\frac{8125 \div 5}{1000 \div 5} = \frac{1625}{200}$

Both 1625 and 200 are still divisible by 5.

$\frac{1625 \div 5}{200 \div 5} = \frac{325}{40}$

Both 325 and 40 are still divisible by 5.

$\frac{325 \div 5}{40 \div 5} = \frac{65}{8}$

Now, check if 65 and 8 have any common factors other than 1.

Prime factors of 65 are $5 \times 13$.

Prime factors of 8 are $2 \times 2 \times 2$.

They do not have any common prime factors, so the fraction $\frac{65}{8}$ is in its lowest terms.

The decimal 8.125 is equal to the fraction $\frac{65}{8}$.

Filling in the blank:

Decimal 8.125 is equal to the fraction $\frac{65}{8}$.

Given:

The subtraction expression: $6.45 – 3.78$

To Find:

The result of the subtraction.

Solution:

We need to subtract 3.78 from 6.45.

Align the decimal points and perform subtraction column by column, starting from the rightmost digit.

Subtract the hundredths place: 5 - 8. We need to borrow from the tenths place.

The 4 in the tenths place becomes 3, and the 5 in the hundredths place becomes 15.

$15 - 8 = 7$. Write down 7 in the hundredths place of the result.

Subtract the tenths place: Now we have 3 - 7. We need to borrow from the units place.

The 6 in the units place becomes 5, and the 3 in the tenths place becomes 13.

$13 - 7 = 6$. Write down 6 in the tenths place of the result.

Place the decimal point in the result, aligned with the decimal points above.

Subtract the units place: Now we have 5 - 3.

$5 - 3 = 2$. Write down 2 in the units place of the result.

The result of the subtraction is 2.67.

We can show the subtraction vertically:

$\begin{array}{cc} & 6 \ . & 4 & 5 \\ - & 3 \ . & 7 & 8 \\ \hline & 2 \ . & 6 & 7 \\ \hline \end{array}$

The value of $6.45 – 3.78$ is 2.67.

Filling in the blank:

Decimal 6.45 – 3.78 = 2.67.

Given:

The statement: The fraction $14\frac{2}{5}$ is equal to 14.2.

To Determine:

Whether the given statement is true or false.

Solution:

To check if the statement is true, we need to convert the mixed fraction $14\frac{2}{5}$ into a decimal and compare it with 14.2.

The mixed fraction $14\frac{2}{5}$ consists of a whole number part (14) and a fractional part ($\frac{2}{5}$).

First, convert the fractional part $\frac{2}{5}$ to a decimal.

To convert a fraction to a decimal, divide the numerator by the denominator.

$\frac{2}{5} = 2 \div 5$

$2 \div 5 = 0.4$

Now, add the decimal value of the fractional part to the whole number part:

$14 + 0.4 = 14.4$

So, the mixed fraction $14\frac{2}{5}$ is equal to the decimal 14.4.

We need to compare this value (14.4) with the given decimal (14.2).

Is $14.4 = 14.2$?

No, $14.4 \neq 14.2$.

Therefore, the statement "The fraction $14\frac{2}{5}$ is equal to 14.2" is false.

The statement is False.

Given:

Two fractions: $\frac{8}{45}$ and $\frac{16}{89}$.

To Compare:

Determine whether $\frac{8}{45}$ is greater than ($>$) or less than ($<$) $\frac{16}{89}$.

Solution:

To compare two fractions, we can use the method of cross-multiplication. For two fractions $\frac{a}{b}$ and $\frac{c}{d}$, if $a \times d > c \times b$, then $\frac{a}{b} > \frac{c}{d}$. If $a \times d < c \times b$, then $\frac{a}{b} < \frac{c}{d}$.

We need to compare $\frac{8}{45}$ and $\frac{16}{89}$.

Here, $a=8$, $b=45$, $c=16$, and $d=89$.

We compare the cross-products $a \times d$ and $c \times b$.

Calculate $8 \times 89$:

$8 \times 89 = 712$

Calculate $16 \times 45$:

$16 \times 45 = 720$

Now, compare the results of the cross-multiplication:

$712$ and $720$.

We see that $712 < 720$.

Since $8 \times 89 < 16 \times 45$, it means that the first fraction is less than the second fraction.

$\frac{8}{45} < \frac{16}{89}$

Filling in the blank:

$\frac{8}{45} \textsf{ < } \frac{16}{89}$

Given:

The fraction $\frac{12}{25}$.

To Convert:

The fraction $\frac{12}{25}$ into a decimal.

Solution:

To convert a fraction to a decimal, we can divide the numerator by the denominator.

$\frac{12}{25} = 12 \div 25$

We perform the division:

Since 12 is less than 25, we start by placing a decimal point and adding a zero to 12, making it 120.

$120 \div 25$. $25 \times 4 = 100$. $25 \times 5 = 125$. So, 25 goes into 120 four times (4 is the first digit after the decimal point).

Subtract $120 - 100 = 20$.

Add another zero to 20, making it 200.

$200 \div 25$. $25 \times 8 = 200$. So, 25 goes into 200 eight times (8 is the second digit after the decimal point).

Subtract $200 - 200 = 0$. The division is complete.

Alternatively, we can convert the denominator to a power of 10. The denominator is 25. We can multiply 25 by 4 to get 100.

Multiply both the numerator and the denominator by 4:

$\frac{12}{25} = \frac{12 \times 4}{25 \times 4}$

$\frac{12 \times 4}{25 \times 4} = \frac{48}{100}$

Now, convert the fraction $\frac{48}{100}$ to a decimal. Dividing by 100 means moving the decimal point two places to the left.

$\frac{48}{100} = 0.48$

Thus, the fraction $\frac{12}{25}$ as a decimal is 0.48.

Given:

Mass in grams: 5809 g.

To Convert:

Convert 5809 g to kilograms (kg).

Solution:

To convert grams (g) to kilograms (kg), we use the conversion factor:

$1 \text{ kg} = 1000 \text{ g}$

This means that 1 gram is equal to $\frac{1}{1000}$ of a kilogram.

$1 \text{ g} = \frac{1}{1000} \text{ kg}$

To convert 5809 g to kilograms, we divide the number of grams by 1000:

$5809 \text{ g} = 5809 \div 1000 \text{ kg}$

Dividing by 1000 is equivalent to moving the decimal point three places to the left.

$5809.0 \div 1000 = 5.809$

So, 5809 g is equal to 5.809 kg.

5809g converted to kg is 5.809 kg.

Given:

The decimal number: 87.952

The place to round off to: tenths place.

To Round Off:

Round off 87.952 to the nearest tenth.

Solution:

To round a decimal to the tenths place, we look at the digit in the hundredths place.

In the number 87.952:

• The units digit is 7.
• The tenths digit is 9.
• The hundredths digit is 5.
• The thousandths digit is 2.

We want to round to the tenths place, so we look at the digit immediately to the right, which is the hundredths digit (5).

The rule for rounding is:

• If the digit in the next place value (here, the hundredths place) is 5 or greater (5, 6, 7, 8, or 9), we round up the digit in the target place value (here, the tenths place).
• If the digit in the next place value is less than 5 (0, 1, 2, 3, or 4), we keep the digit in the target place value as it is (round down).

The digit in the hundredths place is 5.

Since the digit is 5, we round up the digit in the tenths place.

The tenths digit is 9. Rounding up 9 gives 10.

When the tenths digit becomes 10, it means we have 1 whole unit to carry over to the units place.

The 9 in the tenths place becomes 0, and we add 1 to the units place.

The units digit is 7. Adding 1 to 7 gives 8.

The whole number part becomes $87 + 1 = 88$.

The tenths place becomes 0.

So, 87.952 rounded to the tenths place is 88.0.

87.952 rounded off to the tenths place is 88.0.

Given:

The fractions to add: $5\frac{3}{8}$ and $\frac{5}{16}$.

To Find:

The sum of the given fractions.

Solution:

We need to find the sum of $5\frac{3}{8}$ and $\frac{5}{16}$.

First, let's separate the mixed number $5\frac{3}{8}$ into its whole number part and fractional part.

$5\frac{3}{8} = 5 + \frac{3}{8}$

Now, the sum is $(5 + \frac{3}{8}) + \frac{5}{16}$.

We can add the fractional parts together first: $\frac{3}{8} + \frac{5}{16}$.

To add fractions, they must have a common denominator.

The denominators are 8 and 16. The least common multiple (LCM) of 8 and 16 is 16.

Convert $\frac{3}{8}$ to an equivalent fraction with a denominator of 16.

Multiply the numerator and denominator by 2:

$\frac{3}{8} = \frac{3 \times 2}{8 \times 2} = \frac{6}{16}$

Now, add the equivalent fraction $\frac{6}{16}$ to $\frac{5}{16}$:

$\frac{6}{16} + \frac{5}{16} = \frac{6 + 5}{16} = \frac{11}{16}$

Finally, add the whole number part (5) to the sum of the fractional parts ($\frac{11}{16}$).

Sum $= 5 + \frac{11}{16}$

This can be written as a mixed number $5\frac{11}{16}$.

The sum of $5\frac{3}{8}$ and $\frac{5}{16}$ is $5\frac{11}{16}$.

Alternate Solution: Convert to Improper Fractions

Convert the mixed number $5\frac{3}{8}$ to an improper fraction.

$5\frac{3}{8} = \frac{(5 \times 8) + 3}{8} = \frac{40 + 3}{8} = \frac{43}{8}$

Now, add the improper fraction $\frac{43}{8}$ and the fraction $\frac{5}{16}$.

$\frac{43}{8} + \frac{5}{16}$

The common denominator is 16.

Convert $\frac{43}{8}$ to an equivalent fraction with a denominator of 16:

$\frac{43}{8} = \frac{43 \times 2}{8 \times 2} = \frac{86}{16}$

Now, add the fractions with the common denominator:

$\frac{86}{16} + \frac{5}{16} = \frac{86 + 5}{16} = \frac{91}{16}$

The sum as an improper fraction is $\frac{91}{16}$.

We can convert this back to a mixed number by dividing 91 by 16.

$91 \div 16$

$16 \times 5 = 80$

$91 - 80 = 11$ (remainder)

So, $\frac{91}{16} = 5 \text{ with a remainder of } 11 = 5\frac{11}{16}$.

Both methods yield the same result.

The sum of the fractions is $5\frac{11}{16}$.

Given:

Starting number: 37.28

Target sum: 46.8

To Find:

The number that should be added to 37.28 to obtain 46.8.

Solution:

Let the number that should be added be $x$.

According to the problem, the sum of 37.28 and $x$ is 46.8.

To find $x$, we need to isolate it. Subtract 37.28 from both sides of the equation (i):

$x = 46.8 - 37.28$

Perform the subtraction of the decimal numbers.

Align the decimal points. We can write 46.8 as 46.80 to match the number of decimal places in 37.28.

$\begin{array}{cc} & 46 \ . & 8 & 0 \\ - & 37 \ . & 2 & 8 \\ \hline & 9 \ . & 5 & 2 \\ \hline \end{array}$

So, $x = 9.52$.

Verification:

Add the calculated number (9.52) to the starting number (37.28):

$37.28 + 9.52 = 46.80 = 46.8$

This matches the given target sum, 46.8.

The number that should be added to 37.28 to obtain 46.8 is 9.52.

Given:

The decimal numbers: 2.2, 2.023, 2.0226, 22.1, 20.42.

To Arrange:

Arrange the given decimal numbers in ascending order (from smallest to largest).

Solution:

To compare and arrange decimal numbers, it's helpful to make them have the same number of decimal places by adding trailing zeros.

The maximum number of decimal places among the given numbers is four (in 2.0226).

Let's rewrite all the numbers with four decimal places:

• 2.2 becomes 2.2000
• 2.023 becomes 2.0230
• 2.0226 remains 2.0226
• 22.1 becomes 22.1000
• 20.42 becomes 20.4200

The numbers to compare are: 2.2000, 2.0230, 2.0226, 22.1000, 20.4200.

Now, compare the numbers starting from the leftmost digit (the whole number part), then move to the tenths, hundredths, thousandths, and ten-thousandths places.

Compare the whole number parts: 2, 2, 2, 22, 20.

The smallest whole number parts are 2. These are 2.2000, 2.0230, and 2.0226.

The larger whole number parts are 20 and 22. These are 20.4200 and 22.1000. Clearly, 20.4200 is smaller than 22.1000.

Now compare the numbers with whole number part 2: 2.2000, 2.0230, 2.0226.

Compare the tenths place: 2.2000 has 2, 2.0230 has 0, 2.0226 has 0.

So, 2.2000 is larger than the other two.

Now compare 2.0230 and 2.0226.

Compare the hundredths place: Both have 2.

Compare the thousandths place: 2.0230 has 3, 2.0226 has 2.

Since 2 < 3, 2.0226 is smaller than 2.0230.

Ordering the numbers with whole number part 2 from smallest to largest:

2.0226, 2.0230, 2.2000

Ordering all the numbers from smallest to largest:

Start with the smallest group (whole part 2), then the next smallest (whole part 20), then the largest (whole part 22).

2.0226, 2.0230, 2.2000, 20.4200, 22.1000

Rewrite them in their original form:

2.0226, 2.023, 2.2, 20.42, 22.1

The numbers arranged in ascending order are:

2.0226, 2.023, 2.2, 20.42, 22.1

Given:

Weight of apples = 2 kg 280 g

Weight of bananas = 3 kg 375 g

Weight of grapes = 225 g

Weight of oranges = 5 kg 385 g

To Find:

The total weight of the fruits purchased by Gorang in kilograms (kg).

Solution:

To find the total weight in kg, we first need to convert all the weights to kilograms.

We know that $1 \text{ kg} = 1000 \text{ g}$.

So, $1 \text{ g} = \frac{1}{1000} \text{ kg} = 0.001 \text{ kg}$.

Convert each weight to kilograms:

Weight of apples = 2 kg + 280 g $= 2 \text{ kg} + \frac{280}{1000} \text{ kg} = 2 \text{ kg} + 0.280 \text{ kg} = 2.280 \text{ kg}$

Weight of bananas = 3 kg + 375 g $= 3 \text{ kg} + \frac{375}{1000} \text{ kg} = 3 \text{ kg} + 0.375 \text{ kg} = 3.375 \text{ kg}$

Weight of grapes = 225 g $= \frac{225}{1000} \text{ kg} = 0.225 \text{ kg}$

Weight of oranges = 5 kg + 385 g $= 5 \text{ kg} + \frac{385}{1000} \text{ kg} = 5 \text{ kg} + 0.385 \text{ kg} = 5.385 \text{ kg}$

Now, add all the weights in kilograms to find the total weight.

Total weight = Weight of apples + Weight of bananas + Weight of grapes + Weight of oranges

Total weight $= 2.280 \text{ kg} + 3.375 \text{ kg} + 0.225 \text{ kg} + 5.385 \text{ kg}$

Add the decimal numbers:

$\begin{array}{ccccc} & 2 & . & 2 & 8 & 0 \\ & 3 & . & 3 & 7 & 5 \\ & 0 & . & 2 & 2 & 5 \\ + & 5 & . & 3 & 8 & 5 \\ \hline 1 & 1 & . & 2 & 6 & 5 \\ \hline \end{array}$

Starting from the rightmost column (thousandths place):

$0 + 5 + 5 + 5 = 15$. Write down 5, carry over 1.

Next column (hundredths place): $8 + 7 + 2 + 8 + (\text{carry } 1) = 26$. Write down 6, carry over 2.

Next column (tenths place): $2 + 3 + 2 + 3 + (\text{carry } 2) = 12$. Write down 2, carry over 1.

Place the decimal point.

Next column (units place): $2 + 3 + 0 + 5 + (\text{carry } 1) = 11$. Write down 11.

The total weight is 11.265 kg.

The total weight of the fruits purchased by Gorang is 11.265 kg.

Given:

The calculation shown: $\frac{7}{4} + \frac{5}{2} = \frac{7+5}{4+2} = \frac{12}{6} = 2$

To Identify:

What is wrong in the given calculation.

Solution:

The error in the given calculation lies in the step where the fractions are added.

When adding fractions with different denominators, we cannot simply add the numerators together and add the denominators together.

The rule for adding fractions is that they must have a common denominator. Once they have a common denominator, we add the numerators and keep the common denominator the same.

In the given problem, the denominators are 4 and 2. They are different.

The calculation incorrectly adds the numerators $(7+5)$ and the denominators $(4+2)$ independently.

The correct way to add $\frac{7}{4}$ and $\frac{5}{2}$:

Find a common denominator for 4 and 2. The least common multiple (LCM) of 4 and 2 is 4.

Convert $\frac{5}{2}$ to an equivalent fraction with a denominator of 4.

Multiply the numerator and the denominator of $\frac{5}{2}$ by 2:

$\frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4}$

Now, add the fractions $\frac{7}{4}$ and $\frac{10}{4}$ which have the same denominator:

$\frac{7}{4} + \frac{10}{4} = \frac{7 + 10}{4} = \frac{17}{4}$

The correct sum of $\frac{7}{4} + \frac{5}{2}$ is $\frac{17}{4}$.

The value $\frac{17}{4}$ is equal to the mixed number $4\frac{1}{4}$ or the decimal 4.25, which is not equal to 2.

The mistake is in adding the numerators and denominators directly without finding a common denominator.

The step $\frac{7}{4} + \frac{5}{2} = \frac{7+5}{4+2}$ is incorrect.

The given fraction is $\frac{4}{5}$. We need to find which of the given options is not equal to $\frac{4}{5}$.

Let's check each option by simplifying the fraction:

(A) $\frac{40}{50}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $10$.

$\frac{\cancel{40}^{4}}{\cancel{50}_{5}} = \frac{4}{5}$

This fraction is equal to $\frac{4}{5}$.

(B) $\frac{12}{15}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $3$.

$\frac{\cancel{12}^{4}}{\cancel{15}_{5}} = \frac{4}{5}$

This fraction is equal to $\frac{4}{5}$.

(C) $\frac{16}{20}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $4$.

$\frac{\cancel{16}^{4}}{\cancel{20}_{5}} = \frac{4}{5}$

This fraction is equal to $\frac{4}{5}$.

(D) $\frac{9}{15}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $3$.

$\frac{\cancel{9}^{3}}{\cancel{15}_{5}} = \frac{3}{5}$

This fraction is not equal to $\frac{4}{5}$.

Comparing the simplified fractions with $\frac{4}{5}$, we find that options (A), (B), and (C) are equal to $\frac{4}{5}$, while option (D) is not.

Therefore, the fraction which is not equal to $\frac{4}{5}$ is $\frac{9}{15}$.

The correct answer is (D) $\frac{9}{15}$.

We are asked to find the two consecutive integers between which the fraction $\frac{5}{7}$ lies.

To find the integers between which a fraction lies, we can convert the fraction into a decimal number.

Let's convert the fraction $\frac{5}{7}$ into a decimal by dividing the numerator ($5$) by the denominator ($7$).

$\frac{5}{7} \approx 0.714...$

Now, we need to determine which two consecutive integers this decimal number ($0.714...$) lies between.

On the number line, the number $0.714...$ is greater than $0$ and less than $1$.

Therefore, the fraction $\frac{5}{7}$ lies between the integers $0$ and $1$.

Let's check the given options:

• (A) 5 and 6: The fraction $\frac{5}{7} \approx 0.714...$ is not between 5 and 6.
• (B) 0 and 1: The fraction $\frac{5}{7} \approx 0.714...$ is between 0 and 1.
• (C) 5 and 7: The fraction $\frac{5}{7} \approx 0.714...$ is not between 5 and 7.
• (D) 6 and 7: The fraction $\frac{5}{7} \approx 0.714...$ is not between 6 and 7.

The correct option is (B).

The two consecutive integers between which the fraction $\frac{5}{7}$ lies are $0$ and $1$.

The final answer is (B) 0 and 1.

We are given the fraction $\frac{1}{4}$ and asked to find the numerator when this fraction is written with a denominator of $12$.

We want to find an equivalent fraction such that:

$\frac{1}{4} = \frac{\text{New Numerator}}{12}$

To change the denominator from $4$ to $12$, we need to multiply the original denominator by a factor. We can find this factor by dividing the new denominator by the original denominator:

Factor $= \frac{\text{New Denominator}}{\text{Original Denominator}} = \frac{12}{4} = 3$

To maintain the equivalence of the fraction, we must multiply the numerator by the same factor ($3$).

New Numerator $= \text{Original Numerator} \times \text{Factor}$

New Numerator $= 1 \times 3 = 3$

So, the fraction $\frac{1}{4}$ is equivalent to $\frac{3}{12}$.

When $\frac{1}{4}$ is written with a denominator as $12$, its numerator is $3$.

Looking at the options:

• (A) 3
• (B) 8
• (C) 24
• (D) 12

The calculated numerator is $3$, which matches option (A).

The final answer is (A) 3.

A fraction is said to be in its lowest form (or simplest form) if the greatest common divisor (GCD) of its numerator and denominator is $1$. We need to check each option to see if the fraction is in its lowest form.

Let's examine each option:

(A) $\frac{7}{5}$

Numerator = $7$, Denominator = $5$.

The factors of $7$ are $1, 7$.

The factors of $5$ are $1, 5$.

The GCD of $7$ and $5$ is $1$.

So, $\frac{7}{5}$ is in its lowest form.

(B) $\frac{15}{20}$

Numerator = $15$, Denominator = $20$.

The factors of $15$ are $1, 3, 5, 15$.

The factors of $20$ are $1, 2, 4, 5, 10, 20$.

The GCD of $15$ and $20$ is $5$.

Since the GCD is $5$ (which is not $1$), $\frac{15}{20}$ is not in its lowest form. It can be simplified by dividing both numerator and denominator by $5$ to get $\frac{3}{4}$.

(C) $\frac{13}{33}$

Numerator = $13$, Denominator = $33$.

The factors of $13$ are $1, 13$.

The factors of $33$ are $1, 3, 11, 33$.

The GCD of $13$ and $33$ is $1$.

So, $\frac{13}{33}$ is in its lowest form.

(D) $\frac{27}{28}$

Numerator = $27$, Denominator = $28$.

The factors of $27$ are $1, 3, 9, 27$.

The factors of $28$ are $1, 2, 4, 7, 14, 28$.

The GCD of $27$ and $28$ is $1$.

So, $\frac{27}{28}$ is in its lowest form.

From the analysis above, only the fraction in option (B) has a GCD of the numerator and denominator greater than $1$. Therefore, $\frac{15}{20}$ is not in the lowest form.

The final answer is (B) $\frac{15}{20}$.

We are given the equation $\frac{5}{8} = \frac{20}{p}$ and asked to find the value of $p$.

We can solve this proportion by using cross-multiplication. Cross-multiplication states that if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.

Applying this to our equation, we multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction.

Now, to find the value of $p$, we divide both sides of the equation by $5$:

Alternatively, we can observe the relationship between the numerators. The numerator $5$ is multiplied by a factor to get the numerator $20$.

$5 \times \text{Factor} = 20$

Factor $= \frac{20}{5} = 4$

To keep the fractions equivalent, the denominator $8$ must be multiplied by the same factor ($4$) to get $p$.

$p = 8 \times \text{Factor}$

$p = 8 \times 4$

Both methods give the same result for $p$.

Comparing our result with the given options:

• (A) 23
• (B) 2
• (C) 32
• (D) 16

Our calculated value $p=32$ matches option (C).

The final answer is (C) 32.

We need to find which of the given fractions is not equal to the others. To do this, we will simplify each fraction to its lowest form.

(A) $\frac{6}{8}$

The greatest common divisor (GCD) of $6$ and $8$ is $2$. We divide both the numerator and the denominator by $2$.

$\frac{\cancel{6}^{3}}{\cancel{8}_{4}} = \frac{3}{4}$

(B) $\frac{12}{16}$

The greatest common divisor (GCD) of $12$ and $16$ is $4$. We divide both the numerator and the denominator by $4$.

$\frac{\cancel{12}^{3}}{\cancel{16}_{4}} = \frac{3}{4}$

(C) $\frac{15}{25}$

The greatest common divisor (GCD) of $15$ and $25$ is $5$. We divide both the numerator and the denominator by $5$.

$\frac{\cancel{15}^{3}}{\cancel{25}_{5}} = \frac{3}{5}$

(D) $\frac{18}{24}$

The greatest common divisor (GCD) of $18$ and $24$ is $6$. We divide both the numerator and the denominator by $6$.

$\frac{\cancel{18}^{3}}{\cancel{24}_{4}} = \frac{3}{4}$

Comparing the simplified fractions:

• (A) $\frac{3}{4}$
• (B) $\frac{3}{4}$
• (C) $\frac{3}{5}$
• (D) $\frac{3}{4}$

The simplified form of option (C) is $\frac{3}{5}$, while the simplified forms of options (A), (B), and (D) are all $\frac{3}{4}$.

Therefore, the fraction that is not equal to the others is $\frac{15}{25}$.

The final answer is (C) $\frac{15}{25}$.

We are asked to find the greatest fraction among the given options. All the fractions have the same numerator, which is $5$.

When comparing fractions with the same numerator, the fraction with the smaller denominator is the greater fraction. This is because the same total amount (represented by the numerator) is divided into fewer parts (represented by the denominator), making each part larger.

The denominators of the given fractions are:

• (A) $\frac{5}{7}$: Denominator is $7$
• (B) $\frac{5}{6}$: Denominator is $6$
• (C) $\frac{5}{9}$: Denominator is $9$
• (D) $\frac{5}{8}$: Denominator is $8$

We compare the denominators: $7, 6, 9, 8$.

The smallest denominator among these is $6$.

According to the rule, the fraction with the smallest denominator ($6$) will be the greatest. The fraction with denominator $6$ is $\frac{5}{6}$.

Let's list the fractions and their denominators and identify the smallest denominator:

The smallest denominator is $6$, which corresponds to the fraction $\frac{5}{6}$.

Therefore, the greatest fraction is $\frac{5}{6}$.

The correct option is (B).

The final answer is (B) $\frac{5}{6}$.

We are asked to find the smallest fraction among the given options. All the fractions have the same denominator, which is $8$.

When comparing fractions with the same denominator, the fraction with the smaller numerator is the smaller fraction. This is because the whole is divided into the same number of parts, so the fraction with fewer parts taken (smaller numerator) represents a smaller amount.

The numerators of the given fractions are:

• (A) $\frac{7}{8}$: Numerator is $7$
• (B) $\frac{9}{8}$: Numerator is $9$
• (C) $\frac{3}{8}$: Numerator is $3$
• (D) $\frac{5}{8}$: Numerator is $5$

We compare the numerators: $7, 9, 3, 5$.

The smallest numerator among these is $3$.

According to the rule, the fraction with the smallest numerator ($3$) will be the smallest. The fraction with numerator $3$ and denominator $8$ is $\frac{3}{8}$.

Let's list the fractions and their numerators and identify the smallest numerator:

The smallest numerator is $3$, which corresponds to the fraction $\frac{3}{8}$.

Therefore, the smallest fraction is $\frac{3}{8}$.

The correct option is (C).

The final answer is (C) $\frac{3}{8}$.

We are asked to find the sum of the fractions $\frac{4}{17}$ and $\frac{15}{17}$.

The given fractions are $\frac{4}{17}$ and $\frac{15}{17}$. These are like fractions because they have the same denominator, which is $17$.

To add like fractions, we add the numerators and keep the common denominator.

Sum $= \frac{\text{Numerator}_1 + \text{Numerator}_2}{\text{Common Denominator}}$

Sum $= \frac{4 + 15}{17}$

Sum $= \frac{19}{17}$

The sum of $\frac{4}{17}$ and $\frac{15}{17}$ is $\frac{19}{17}$.

Let's check the options:

• (A) $\frac{19}{17}$
• (B) $\frac{11}{17}$
• (C) $\frac{19}{34}$
• (D) $\frac{2}{17}$

Our result matches option (A).

The final answer is (A) $\frac{19}{17}$.

We are asked to subtract $\frac{5}{9}$ from $\frac{19}{9}$. This means we need to calculate $\frac{19}{9} - \frac{5}{9}$.

The given fractions are $\frac{19}{9}$ and $\frac{5}{9}$. These are like fractions because they have the same denominator, which is $9$.

To subtract like fractions, we subtract the numerators and keep the common denominator.

Difference $= \frac{\text{Numerator}_1 - \text{Numerator}_2}{\text{Common Denominator}}$

Difference $= \frac{19 - 5}{9}$

Difference $= \frac{14}{9}$

The result of subtracting $\frac{5}{9}$ from $\frac{19}{9}$ is $\frac{14}{9}$.

Let's check the options:

• (A) $\frac{24}{9}$
• (B) $\frac{14}{9}$
• (C) $\frac{14}{18}$
• (D) $\frac{14}{0}$

Our result matches option (B).

The final answer is (B) $\frac{14}{9}$.

We are asked to find the two consecutive numbers between which the decimal number $0.7499$ lies.

To determine the interval, we compare the given number $0.7499$ with the lower and upper bounds of each option.

Let's examine each option:

(A) 0.7 and 0.74

We need to check if $0.7 < 0.7499 < 0.74$.

Comparing $0.7$ and $0.7499$: $0.7000 < 0.7499$. This part is true.

Comparing $0.7499$ and $0.74$: $0.7499 > 0.7400$. This part is false.

So, $0.7499$ does not lie between $0.7$ and $0.74$.

(B) 0.75 and 0.79

We need to check if $0.75 < 0.7499 < 0.79$.

Comparing $0.75$ and $0.7499$: $0.7500 > 0.7499$. This part is false.

So, $0.7499$ does not lie between $0.75$ and $0.79$.

(C) 0.749 and 0.75

We need to check if $0.749 < 0.7499 < 0.75$.

Comparing $0.749$ and $0.7499$: $0.7490 < 0.7499$. This part is true.

Comparing $0.7499$ and $0.75$: $0.7499 < 0.7500$. This part is true.

Since both conditions are met, $0.7499$ lies between $0.749$ and $0.75$.

(D) 0.74992 and 0.75

We need to check if $0.74992 < 0.7499 < 0.75$.

Comparing $0.74992$ and $0.7499$: $0.74992 > 0.74990$. This part is false.

So, $0.7499$ does not lie between $0.74992$ and $0.75$.

Based on the comparisons, the number $0.7499$ lies between $0.749$ and $0.75$.

The correct option is (C).

The final answer is (C) 0.749 and 0.75.

We are asked to find the two consecutive numbers between which the decimal number $0.023$ lies.

To determine the interval, we need to compare the given number $0.023$ with the lower and upper bounds provided in each option.

Let's examine each option:

(A) 0.2 and 0.3

We check if $0.2 < 0.023 < 0.3$.

Comparing $0.2$ and $0.023$: $0.200 > 0.023$.

Since $0.2$ is greater than $0.023$, the condition $0.2 < 0.023$ is false.

So, $0.023$ does not lie between $0.2$ and $0.3$.

(B) 0.02 and 0.03

We check if $0.02 < 0.023 < 0.03$.

Comparing $0.02$ and $0.023$: $0.020 < 0.023$. This is true.

Comparing $0.023$ and $0.03$: $0.023 < 0.030$. This is true.

Since both conditions are met, $0.023$ lies between $0.02$ and $0.03$.

(C) 0.03 and 0.029

The numbers are given as 0.03 and 0.029. Note that $0.029 < 0.03$. We check if $0.029 < 0.023 < 0.03$.

Comparing $0.029$ and $0.023$: $0.029 > 0.023$.

Since $0.029$ is greater than $0.023$, the condition $0.029 < 0.023$ is false.

So, $0.023$ does not lie between $0.03$ and $0.029$ (or $0.029$ and $0.03$).

(D) 0.026 and 0.024

The numbers are given as 0.026 and 0.024. Note that $0.024 < 0.026$. We check if $0.024 < 0.023 < 0.026$.

Comparing $0.024$ and $0.023$: $0.024 > 0.023$.

Since $0.024$ is greater than $0.023$, the condition $0.024 < 0.023$ is false.

So, $0.023$ does not lie between $0.026$ and $0.024$ (or $0.024$ and $0.026$).

Based on the comparisons, the number $0.023$ lies between $0.02$ and $0.03$.

The correct option is (B).

The final answer is (B) 0.02 and 0.03.

We are asked to express the improper fraction $\frac{11}{7}$ in mixed number form.

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert an improper fraction to a mixed number, we divide the numerator by the denominator.

Divide $11$ by $7$:

$11 \div 7$

We find the quotient and the remainder:

$11 = 7 \times 1 + 4$

Here, the quotient is $1$, and the remainder is $4$. The denominator remains $7$.

The mixed number is formed as: Quotient $\frac{\text{Remainder}}{\text{Denominator}}$.

So, $\frac{11}{7} = 1 \frac{4}{7}$.

Let's check the given options:

• (A) $7\frac{1}{4} = \frac{7 \times 4 + 1}{4} = \frac{28 + 1}{4} = \frac{29}{4}$
• (B) $4\frac{1}{7} = \frac{4 \times 7 + 1}{7} = \frac{28 + 1}{7} = \frac{29}{7}$
• (C) $1\frac{4}{7} = \frac{1 \times 7 + 4}{7} = \frac{7 + 4}{7} = \frac{11}{7}$
• (D) $11\frac{1}{7} = \frac{11 \times 7 + 1}{7} = \frac{77 + 1}{7} = \frac{78}{7}$

Comparing our result with the options, we see that $1\frac{4}{7}$ is equal to $\frac{11}{7}$.

The correct option is (C).

The final answer is (C) $1\frac{4}{7}$.

We are asked to express the mixed fraction $5\frac{4}{7}$ as an improper fraction.

A mixed fraction consists of a whole number and a proper fraction. To convert a mixed fraction to an improper fraction, we follow these steps:

1. Multiply the whole number by the denominator of the fractional part.

2. Add the numerator of the fractional part to the result from step 1.

3. The result from step 2 becomes the numerator of the improper fraction, and the denominator remains the same as the denominator of the original fractional part.

For the mixed fraction $5\frac{4}{7}$:

Whole number = $5$

Numerator of fractional part = $4$}

Denominator of fractional part = $7$

Step 1: Multiply the whole number by the denominator:

$5 \times 7 = 35$

Step 2: Add the numerator of the fractional part to the result from Step 1:

$35 + 4 = 39$

This is the new numerator.

Step 3: Keep the same denominator:

The denominator is $7$.

So, the improper fraction is $\frac{39}{7}$.

$5\frac{4}{7} = \frac{5 \times 7 + 4}{7} = \frac{35 + 4}{7} = \frac{39}{7}$

Let's check the given options:

• (A) $\frac{33}{7}$
• (B) $\frac{39}{7}$
• (C) $\frac{33}{4}$
• (D) $\frac{39}{4}$

Our result matches option (B).

The final answer is (B) $\frac{39}{7}$.

We are asked to find the sum of $0.07$ and $0.008$.

To add decimal numbers, we need to align the decimal points and then add the digits in each place value column, starting from the rightmost digit.

We can write $0.07$ as $0.070$ to have the same number of decimal places as $0.008$.

Adding $0.070$ and $0.008$:

$0.07 + 0.008 = 0.078$

Let's check the given options:

• (A) $0.15$
• (B) $0.015$
• (C) $0.078$
• (D) $0.78$

Our calculated sum is $0.078$, which matches option (C).

The final answer is (C) $0.078$.

We are asked to find the greatest decimal among the given options.

To compare decimals, we compare the digits from left to right, starting from the whole number part (which is $0$ for all options). If the whole numbers are the same, we compare the digits in the tenths place, then the hundredths place, and so on.

Let's write all the decimals with the same number of decimal places (up to the fourth decimal place, as in option B):

• (A) $0.1820$
• (B) $0.0925$
• (C) $0.2900$
• (D) $0.0380$

Now, let's compare the digits place by place:

Compare the tenths place:

• (A) $0.\underline{1}820$ - digit is $1$
• (B) $0.\underline{0}925$ - digit is $0$
• (C) $0.\underline{2}900$ - digit is $2$
• (D) $0.\underline{0}380$ - digit is $0$

The largest digit in the tenths place is $2$ (option C). Therefore, $0.2900$ is the greatest decimal.

We can also compare directly without adding trailing zeros:

• $0.182$
• $0.0925$
• $0.29$
• $0.038$

The tenths digits are $1, 0, 2, 0$. The largest tenths digit is $2$, which corresponds to $0.29$.

Therefore, the greatest decimal is $0.29$.

The correct option is (C).

The final answer is (C) 0.29.

We are asked to find the smallest decimal among the given options.

To compare decimals, we first compare the whole number parts. If the whole number parts are the same, we compare the digits in the tenths place, then the hundredths place, and so on, from left to right.

The given decimals are:

• (A) $0.27$
• (B) $1.5$
• (C) $0.082$
• (D) $0.103$

Let's compare the whole number parts:

• (A) $\underline{0}.27$ - whole number part is $0$
• (B) $\underline{1}.5$ - whole number part is $1$
• (C) $\underline{0}.082$ - whole number part is $0$
• (D) $\underline{0}.103$ - whole number part is $0$

Option (B) has a whole number part of $1$, while the others have $0$. Since $1 > 0$, $1.5$ is greater than the other three decimals. We are looking for the smallest, so $1.5$ cannot be the smallest.

Now we compare the decimals with a whole number part of $0$: $0.27$, $0.082$, and $0.103$.

Let's compare the digits in the tenths place:

• (A) $0.\underline{2}7$ - tenths digit is $2$
• (C) $0.\underline{0}82$ - tenths digit is $0$
• (D) $0.\underline{1}03$ - tenths digit is $1$

The tenths digits are $2, 0, 1$. The smallest tenths digit is $0$, which belongs to $0.082$.

Since the tenths digit of $0.082$ ($0$) is smaller than the tenths digits of $0.27$ ($2$) and $0.103$ ($1$), $0.082$ is the smallest among $0.27$, $0.082$, and $0.103$.

Therefore, the smallest decimal among all the options is $0.082$.

The correct option is (C).

The final answer is (C) 0.082.

We are asked to round the decimal number $13.572$ to the tenths place.

The tenths place is the first digit to the right of the decimal point. In the number $13.572$, the digit in the tenths place is $5$.

$13.\underline{5}72$

To round to the tenths place, we look at the digit immediately to the right of the tenths place, which is the digit in the hundredths place. In $13.572$, the digit in the hundredths place is $7$.

$13.5\underline{7}2$

The rule for rounding is:

• If the digit to the right of the rounding place is $5$ or greater, we round up the digit in the rounding place.
• If the digit to the right of the rounding place is less than $5$, we keep the digit in the rounding place as it is.

In our case, the digit in the hundredths place is $7$, which is greater than or equal to $5$. Therefore, we round up the digit in the tenths place ($5$).

Rounding up $5$ gives us $6$.

So, $13.572$ rounded to the tenths place is $13.6$. The digits to the right of the tenths place are dropped.

Let's check the given options:

• (A) 10
• (B) 13.57
• (C) 14.5
• (D) 13.6

Our result matches option (D).

The final answer is (D) 13.6.

We are asked to find the result of subtracting $6.73$ from $15.8$. This means we need to calculate $15.8 - 6.73$.

To subtract decimal numbers, we need to align the decimal points and then subtract the digits in each place value column, starting from the rightmost digit. We can add trailing zeros to the number with fewer decimal places to make the subtraction easier.

We can write $15.8$ as $15.80$ to have the same number of decimal places as $6.73$.

Subtracting $6.73$ from $15.80$:

$15.80 - 6.73 = 9.07$

Let's perform the subtraction column by column:

• In the hundredths column, we have $0 - 3$. We need to borrow from the tenths place. The $8$ in the tenths place becomes $7$, and the $0$ in the hundredths place becomes $10$. So, $10 - 3 = 7$.
• In the tenths column, we now have $7 - 7 = 0$.
• Place the decimal point.
• In the units column, we have $5 - 6$. We need to borrow from the tens place. The $1$ in the tens place becomes $0$, and the $5$ in the units place becomes $15$. So, $15 - 6 = 9$.
• In the tens column, we have $0 - 0 = 0$.

The result is $9.07$.

Let's check the given options:

• (A) 8.07
• (B) 9.07
• (C) 9.13
• (D) 9.25

Our calculated difference is $9.07$, which matches option (B).

The final answer is (B) 9.07.

We are asked to convert the decimal number $0.238$ into a fraction.

To convert a decimal to a fraction, we look at the number of decimal places. The number $0.238$ has three decimal places. This means the denominator of the initial fraction will be $1000$ (since $10^3 = 1000$). The digits after the decimal point ($238$) will form the numerator.

$0.238 = \frac{238}{1000}$

Now, we need to simplify this fraction to its lowest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).

The numerator is $238$ and the denominator is $1000$. Both numbers are even, so they are divisible by $2$.

Divide the numerator by $2$:

$238 \div 2 = 119$

Divide the denominator by $2$:

$1000 \div 2 = 500$}

So, $\frac{238}{1000} = \frac{119}{500}$.

Now, we check if the fraction $\frac{119}{500}$ can be simplified further. We need to find the GCD of $119$ and $500$.

Let's find the prime factors of $119$ and $500$.

Prime factors of $119$: $119 = 7 \times 17$.

Prime factors of $500$: $500 = 5 \times 100 = 5 \times 10 \times 10 = 5 \times (2 \times 5) \times (2 \times 5) = 2^2 \times 5^3 = 4 \times 125$.

The prime factors of $119$ are $7$ and $17$.

The prime factors of $500$ are $2$ and $5$.

There are no common prime factors other than $1$. Therefore, the GCD of $119$ and $500$ is $1$.

The fraction $\frac{119}{500}$ is in its lowest form.

Let's check the given options:

• (A) $\frac{119}{500}$
• (B) $\frac{238}{25}$
• (C) $\frac{119}{25}$
• (D) $\frac{119}{50}$

Our simplified fraction $\frac{119}{500}$ matches option (A).

The final answer is (A) $\frac{119}{500}$.

The completed statement is: A number representing a part of a whole is called a fraction.

A fraction is a mathematical expression used to denote a quantity that is a part of a specific unit or collection, which is termed the whole.

The "whole" refers to the entire object or quantity that is being divided into equal parts. This whole can be a single item, a group of items, or even a single unit in measurement.

For instance, if you have one apple (the whole) and you cut it into two equal pieces, each piece represents $\frac{1}{2}$ of the whole apple. The fraction $\frac{1}{2}$ represents a part of the whole.

Another example: If there are 5 pens in a box (the whole), and you take out 2 pens, you have taken a part of the whole. This part can be represented by the fraction $\frac{2}{5}$.

In general, a fraction is written in the form $\frac{\text{Numerator}}{\text{Denominator}}$, where the denominator indicates the total number of equal parts the whole is divided into, and the numerator indicates how many of those equal parts are being considered. Thus, a fraction specifically represents a number that is a fraction of a complete unit or set, which is the whole.

A fraction with denominator greater than the numerator is called a proper fraction.

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number).

In mathematical terms, for a proper fraction $\frac{a}{b}$, the condition is $a < b$, where $a$ and $b$ are positive integers.

Proper fractions represent quantities that are less than one whole. For example, if a pizza is divided into 8 equal slices, taking 3 slices represents the fraction $\frac{3}{8}$. Here, the numerator (3) is less than the denominator (8), and the quantity (3 slices) is less than the whole pizza (8 slices).

Examples of proper fractions include $\frac{1}{2}$, $\frac{3}{4}$, $\frac{5}{6}$, $\frac{7}{100}$, etc. In each case, the numerator is smaller than the denominator.

Fractions with the same denominator are called like fractions.

Like fractions are fractions that share the same denominator. The denominator represents the total number of equal parts into which a whole is divided. When fractions have the same denominator, it means they are expressing parts of wholes that have been divided into the same number of equal pieces.

For example, the fractions $\frac{1}{8}$, $\frac{3}{8}$, $\frac{5}{8}$, and $\frac{7}{8}$ are like fractions because they all have the same denominator, which is 8. This means that each fraction represents a certain number of 'eighths' of a whole.

Working with like fractions is simpler than working with unlike fractions (those with different denominators), especially when performing addition or subtraction. Since the size of the parts is the same (determined by the common denominator), you only need to add or subtract the numerators.

$13\frac{5}{18}$ is a mixed fraction.

A mixed fraction (or mixed number) is a combination of a whole number and a proper fraction.

In the given example, $13\frac{5}{18}$, the whole number part is $13$ and the proper fraction part is $\frac{5}{18}$. A proper fraction is one where the numerator (5) is less than the denominator (18).

Mixed fractions are typically used to represent quantities greater than one whole. They can be converted into improper fractions (where the numerator is greater than or equal to the denominator) and vice versa.

To convert $13\frac{5}{18}$ to an improper fraction, you multiply the whole number by the denominator of the fraction and add the numerator. The result becomes the new numerator, and the denominator remains the same:

Improper fraction = $\frac{(\text{Whole Number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}$

Improper fraction = $\frac{(13 \times 18) + 5}{18}$

Improper fraction = $\frac{234 + 5}{18}$

Improper fraction = $\frac{239}{18}$

Thus, $13\frac{5}{18}$ is a mixed fraction, equivalent to the improper fraction $\frac{239}{18}$.

$\frac{18}{5}$ is an improper fraction.

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

In the fraction $\frac{18}{5}$, the numerator is 18 and the denominator is 5. Since $18 > 5$, the numerator is greater than the denominator. This fits the definition of an improper fraction.

Improper fractions represent a quantity that is one whole or more than one whole. They can be converted into mixed numbers (a whole number plus a proper fraction) or a whole number if the numerator is a multiple of the denominator.

To convert the improper fraction $\frac{18}{5}$ into a mixed number, you divide the numerator by the denominator:

$18 \div 5$

The quotient is 3 and the remainder is 3.

So, $\frac{18}{5}$ can be written as the mixed number $3\frac{3}{5}$. Here, 3 is the whole number part, and $\frac{3}{5}$ is the proper fraction part.

Examples of improper fractions include $\frac{5}{2}$, $\frac{7}{7}$, $\frac{10}{3}$, $\frac{25}{4}$, etc. In each case, the numerator is greater than or equal to the denominator.

$\frac{7}{19}$ is a proper fraction.

A proper fraction is defined as a fraction where the numerator (the top number) is less than the denominator (the bottom number).

In the given fraction $\frac{7}{19}$, the numerator is $7$ and the denominator is $19$. Since $7 < 19$, the numerator is less than the denominator.

Proper fractions represent quantities that are strictly less than one whole. For instance, if a whole is divided into 19 equal parts, taking 7 of those parts represents a portion that is less than the entire whole. Therefore, $\frac{7}{19}$ represents a value less than 1.

$\frac{5}{8}$ and $\frac{3}{8}$ are like proper fractions.

The term that describes fractions with the same denominator is "like". Both fractions $\frac{5}{8}$ and $\frac{3}{8}$ have the same denominator, which is 8. Therefore, they are like fractions.

Additionally, both fractions are proper fractions because their numerators are less than their denominators. For $\frac{5}{8}$, the numerator $5$ is less than the denominator $8$ ($5 < 8$). For $\frac{3}{8}$, the numerator $3$ is less than the denominator $8$ ($3 < 8$).

Since both fractions satisfy the criteria for being "like" (same denominator) and "proper" (numerator less than denominator), the most appropriate term to fill the blank describing both properties simultaneously is "like".

$\frac{6}{11}$ and $\frac{6}{13}$ are unlike proper fractions.

The fractions $\frac{6}{11}$ and $\frac{6}{13}$ have different denominators (11 and 13). Fractions with different denominators are called unlike fractions.

Both fractions are also proper fractions because in each case, the numerator is less than the denominator:

$6 < 11$ for the fraction $\frac{6}{11}$.

$6 < 13$ for the fraction $\frac{6}{13}$.

Therefore, the fractions $\frac{6}{11}$ and $\frac{6}{13}$ are unlike proper fractions.

The fraction $\frac{6}{15}$ in simplest form is $\frac{2}{5}$.

To express a fraction in its simplest form (also known as reduced form), we need to divide both the numerator and the denominator by their greatest common divisor (GCD).

The given fraction is $\frac{6}{15}$.

Numerator = 6

Denominator = 15

We find the factors of the numerator and the denominator:

Factors of 6 are 1, 2, 3, 6.

Factors of 15 are 1, 3, 5, 15.

The common factors are 1 and 3.

The greatest common divisor (GCD) of 6 and 15 is 3.

Now, we divide both the numerator and the denominator by the GCD (3):

New Numerator = $6 \div 3 = 2$

New Denominator = $15 \div 3 = 5$

The simplified fraction is $\frac{2}{5}$.

We can also show the cancellation:

$\frac{\cancel{6}^2}{\cancel{15}_5} = \frac{2}{5}$

Since the only common factor of 2 and 5 is 1, the fraction $\frac{2}{5}$ is in its simplest form.

The fraction $\frac{17}{34}$ in simplest form is $\frac{1}{2}$.

To write a fraction in its simplest form, we need to divide both the numerator and the denominator by their greatest common divisor (GCD).

The given fraction is $\frac{17}{34}$.

Numerator = 17

Denominator = 34

We find the factors of the numerator and the denominator:

Factors of 17 are 1, 17.

Factors of 34 are 1, 2, 17, 34.

The common factors of 17 and 34 are 1 and 17.

The greatest common divisor (GCD) of 17 and 34 is 17.

Now, we divide both the numerator and the denominator by the GCD (17):

New Numerator = $17 \div 17 = 1$

New Denominator = $34 \div 17 = 2$

The simplified fraction is $\frac{1}{2}$.

We can show the cancellation process:

$\frac{\cancel{17}^1}{\cancel{34}_2} = \frac{1}{2}$

Since the only common factor of 1 and 2 is 1, the fraction $\frac{1}{2}$ is in its simplest form.

$\frac{18}{135}$ and $\frac{90}{675}$ are proper, unlike and equivalent fractions.

Let's verify the properties mentioned and find the missing one:

1. Proper Fractions:

A fraction is proper if the numerator is less than the denominator.

For $\frac{18}{135}$: $18 < 135$. This is true, so it is a proper fraction.

For $\frac{90}{675}$: $90 < 675$. This is true, so it is a proper fraction.

2. Unlike Fractions:

Fractions are unlike if they have different denominators.

The denominator of the first fraction is 135.

The denominator of the second fraction is 675.

Since $135 \neq 675$, the fractions are unlike.

3. Missing Property:

We need to find another property that applies to both fractions. Let's check if they represent the same value by simplifying them to their simplest form.

Simplify $\frac{18}{135}$:

We can find the greatest common divisor (GCD) of 18 and 135. Or we can divide by common factors successively.

Both are divisible by 3:

$\frac{18 \div 3}{135 \div 3} = \frac{6}{45}$

Both are still divisible by 3:

$\frac{6 \div 3}{45 \div 3} = \frac{2}{15}$

The fraction $\frac{2}{15}$ is in simplest form as the only common factor of 2 and 15 is 1.

Alternatively using prime factorization:

$18 = 2 \times 3^2$

$135 = 3^3 \times 5$

GCD(18, 135) = $3^2 = 9$

$\frac{18 \div 9}{135 \div 9} = \frac{2}{15}$

Simplify $\frac{90}{675}$:

Both are divisible by 5:

$\frac{90 \div 5}{675 \div 5} = \frac{18}{135}$

We already simplified $\frac{18}{135}$ to $\frac{2}{15}$.

Alternatively using prime factorization:

$90 = 2 \times 3^2 \times 5$

$675 = 3^3 \times 5^2$

GCD(90, 675) = $3^2 \times 5 = 9 \times 5 = 45$

$\frac{90 \div 45}{675 \div 45} = \frac{2}{15}$

Since both fractions $\frac{18}{135}$ and $\frac{90}{675}$ simplify to the same fraction $\frac{2}{15}$, they represent the same value. Fractions that represent the same value are called equivalent fractions.

$8\frac{2}{7}$ is equal to the improper fraction $\frac{58}{7}$.

$8\frac{2}{7}$ is a mixed number, which consists of a whole number part (8) and a proper fraction part ($\frac{2}{7}$). An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

To convert a mixed number like $8\frac{2}{7}$ into an improper fraction, we follow these steps:

Step 1: Multiply the whole number part by the denominator of the fraction.

$8 \times 7 = 56$

Step 2: Add the numerator of the fraction to the result from Step 1. This sum becomes the new numerator of the improper fraction.

$56 + 2 = 58$

Step 3: Keep the original denominator of the fraction as the denominator of the improper fraction.

The denominator is 7.

So, the improper fraction equivalent to $8\frac{2}{7}$ is $\frac{58}{7}$.

$\frac{87}{7}$ is equal to the mixed fraction $12\frac{3}{7}$.

$\frac{87}{7}$ is an improper fraction because the numerator (87) is greater than the denominator (7). A mixed fraction consists of a whole number and a proper fraction.

To convert an improper fraction to a mixed fraction, we perform division.

We divide the numerator (87) by the denominator (7):

$87 \div 7$

We perform the division:

$\begin{array}{r} 12\phantom{)} \\ 7{\overline{\smash{\big)}\,87\phantom{)}}} \\ \underline{-~\phantom{(}7\phantom{).}} \\ 17\phantom{)} \\ \underline{-~14\phantom{)}} \\ 3\phantom{)} \end{array}$

The result of the division gives us:

Quotient = 12

Remainder = 3

Divisor = 7

To form the mixed fraction, the quotient becomes the whole number part, the remainder becomes the numerator of the fractional part, and the divisor remains the denominator:

Mixed Fraction = $\text{Quotient} \frac{\text{Remainder}}{\text{Divisor}}$

Mixed Fraction = $12 \frac{3}{7}$

So, the improper fraction $\frac{87}{7}$ is equal to the mixed fraction $12\frac{3}{7}$.

$9 + \frac{2}{10} + \frac{6}{100}$ is equal to the decimal number 9.26.

The given expression is a sum of a whole number and two fractions:

$9 + \frac{2}{10} + \frac{6}{100}$

To convert this sum into a decimal number, we need to understand the place value system in decimals.

The fraction $\frac{2}{10}$ represents two tenths. In the decimal system, the first digit to the right of the decimal point represents the tenths place.

So, $\frac{2}{10} = 0.2$

The fraction $\frac{6}{100}$ represents six hundredths. The second digit to the right of the decimal point represents the hundredths place.

So, $\frac{6}{100} = 0.06$

Now we can rewrite the original expression using the decimal equivalents of the fractions:

$9 + 0.2 + 0.06$

Adding these values together:

The whole number part is 9.

The tenths part is 0.2.

The hundredths part is 0.06.

We can align the decimal points and add:

$\begin{array}{cc} & 9.00 \\ + & 0.20 \\ + & 0.06 \\ \hline & 9.26 \\ \hline \end{array}$

Thus, $9 + \frac{2}{10} + \frac{6}{100} = 9.26$.

Decimal 16.25 is equal to the fraction $16\frac{1}{4}$ (or $\frac{65}{4}$).

To convert a decimal number to a fraction, we consider the place value of the digits after the decimal point.

The decimal number is 16.25.

This number can be broken down into a whole number part and a decimal part:

$16.25 = 16 + 0.25$

Now, let's convert the decimal part (0.25) into a fraction.

The digit '2' is in the tenths place, and the digit '5' is in the hundredths place. This means 0.25 can be read as "twenty-five hundredths".

As a fraction, this is written as $\frac{25}{100}$.

So, we have $16 + \frac{25}{100}$. This can be written as a mixed number $16\frac{25}{100}$.

To express the fraction in its simplest form, we need to simplify the fractional part $\frac{25}{100}$.

We find the greatest common divisor (GCD) of the numerator 25 and the denominator 100.

Factors of 25: 1, 5, 25

Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

The GCD of 25 and 100 is 25.

Divide both the numerator and the denominator by 25:

$\frac{25 \div 25}{100 \div 25} = \frac{1}{4}$

Using cancellation:

$\frac{\cancel{25}^1}{\cancel{100}_4} = \frac{1}{4}$

So, the mixed number in simplest form is $16\frac{1}{4}$.

If an improper fraction is required, we convert the mixed number $16\frac{1}{4}$:

Improper fraction = $\frac{(\text{Whole Number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}$

= $\frac{(16 \times 4) + 1}{4}$

= $\frac{64 + 1}{4}$

= $\frac{65}{4}$

Both $16\frac{1}{4}$ and $\frac{65}{4}$ are valid fractional representations of 16.25, with $16\frac{1}{4}$ being the mixed number form in simplest terms.

Fraction $\frac{7}{25}$ is equal to the decimal number 0.28.

To convert a fraction into a decimal, we can either perform division (numerator divided by denominator) or find an equivalent fraction with a denominator that is a power of 10 (such as 10, 100, 1000, etc.).

Let's use the method of finding an equivalent fraction with a denominator as a power of 10.

The denominator of the fraction $\frac{7}{25}$ is 25. We can easily convert 25 into 100 by multiplying it by 4.

To keep the fraction equivalent, we must multiply both the numerator and the denominator by the same number (4):

$\frac{7}{25} = \frac{7 \times 4}{25 \times 4} = \frac{28}{100}$

The equivalent fraction is $\frac{28}{100}$.

The fraction $\frac{28}{100}$ means "28 hundredths". In decimal notation, the first digit after the decimal point is the tenths place, and the second digit is the hundredths place.

So, $\frac{28}{100}$ is written as 0.28.

Alternatively, using division:

We divide 7 by 25.

$\begin{array}{r} 0.28 \\ 25{\overline{\smash{\big)}\,7.00}} \\ \underline{-~\phantom{(}5\,0\phantom{).}} \\ 2\,00\phantom{)} \\ \underline{-~2\,00\phantom{)}} \\ 0\phantom{)} \end{array}$

The result of the division is 0.28.

$\frac{17}{9} + \frac{41}{9} = \frac{58}{9}$.

The given problem is the addition of two fractions: $\frac{17}{9}$ and $\frac{41}{9}$.

These are like fractions because they share the same denominator, which is 9.

To add like fractions, we add the numerators and keep the denominator the same.

Sum of numerators = Numerator 1 + Numerator 2

= $17 + 41$

$\begin{array}{cc} & 1 & 7 \\ + & 4 & 1 \\ \hline & 5 & 8 \\ \hline \end{array}$

Sum of numerators = 58

The denominator remains 9.

So, the sum of the fractions is $\frac{58}{9}$.

$\frac{17}{9} + \frac{41}{9} = \frac{17 + 41}{9} = \frac{58}{9}$.

The resulting fraction $\frac{58}{9}$ is an improper fraction (since $58 > 9$). It can also be expressed as a mixed number by dividing the numerator by the denominator:

$58 \div 9$ gives a quotient of 6 and a remainder of 4.

Thus, $\frac{58}{9} = 6\frac{4}{9}$.

$\frac{67}{14} - \frac{24}{14} = \frac{43}{14}$.

The problem asks us to find the difference between two fractions: $\frac{67}{14}$ and $\frac{24}{14}$.

These fractions are like fractions because they have the same denominator, which is 14.

To subtract like fractions, we subtract the numerator of the second fraction from the numerator of the first fraction and keep the denominator the same.

Difference of numerators = Numerator 1 - Numerator 2

= $67 - 24$

$\begin{array}{cc} & 6 & 7 \\ - & 2 & 4 \\ \hline & 4 & 3 \\ \hline \end{array}$

Difference of numerators = 43

The denominator remains 14.

So, the result of the subtraction is $\frac{43}{14}$.

$\frac{67}{14} - \frac{24}{14} = \frac{67 - 24}{14} = \frac{43}{14}$.

The resulting fraction $\frac{43}{14}$ is an improper fraction since the numerator (43) is greater than the denominator (14). It can also be written as a mixed number.

To convert $\frac{43}{14}$ to a mixed number, divide 43 by 14:

$43 \div 14$ gives a quotient of 3 and a remainder of 1.

So, $\frac{43}{14} = 3\frac{1}{14}$.

$\frac{17}{2} + 3\frac{1}{2} = 12$.

To add the improper fraction $\frac{17}{2}$ and the mixed number $3\frac{1}{2}$, we first need to convert the mixed number into an improper fraction.

Convert $3\frac{1}{2}$ to an improper fraction:

Multiply the whole number (3) by the denominator (2), and then add the numerator (1).

New numerator = $(3 \times 2) + 1 = 6 + 1 = 7$

The denominator remains the same (2).

So, $3\frac{1}{2} = \frac{7}{2}$.

Now the problem becomes the addition of two improper fractions:

$\frac{17}{2} + \frac{7}{2}$

These are like fractions because they have the same denominator (2).

To add like fractions, we add the numerators and keep the denominator the same:

Sum of numerators = $17 + 7$

$\begin{array}{cc} & 1 & 7 \\ + & & 7 \\ \hline & 2 & 4 \\ \hline \end{array}$

Sum of numerators = 24

The resulting fraction is $\frac{24}{2}$.

$\frac{17}{2} + \frac{7}{2} = \frac{17 + 7}{2} = \frac{24}{2}$.

The fraction $\frac{24}{2}$ can be simplified by dividing the numerator by the denominator:

$\frac{24}{2} = 24 \div 2 = 12$.

Thus, $\frac{17}{2} + 3\frac{1}{2} = 12$.

$9\frac{1}{4} + \frac{5}{4} = 10\frac{1}{2}$ (or $\frac{21}{2}$).

To add a mixed number and a fraction, we first convert the mixed number into an improper fraction.

Convert the mixed number $9\frac{1}{4}$ to an improper fraction:

Multiply the whole number (9) by the denominator (4) and add the numerator (1).

New numerator = $(9 \times 4) + 1 = 36 + 1 = 37$.

The denominator remains the same, which is 4.

So, $9\frac{1}{4}$ is equivalent to the improper fraction $\frac{37}{4}$.

Now, the problem is to add the two fractions:

$\frac{37}{4} + \frac{5}{4}$

These are like fractions because they have the same denominator (4).

To add like fractions, we add the numerators and keep the denominator the same:

Sum of numerators = $37 + 5 = 42$.

The denominator is 4.

So, the sum is $\frac{42}{4}$.

$\frac{37}{4} + \frac{5}{4} = \frac{37 + 5}{4} = \frac{42}{4}$.

The resulting fraction $\frac{42}{4}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

$\frac{42 \div 2}{4 \div 2} = \frac{21}{2}$

Using cancellation:

$\frac{\cancel{42}^{21}}{\cancel{4}_2} = \frac{21}{2}$

The improper fraction $\frac{21}{2}$ can also be written as a mixed number.

Divide 21 by 2:

$\begin{array}{r} 10\phantom{)} \\ 2{\overline{\smash{\big)}\,21\phantom{)}}} \\ \underline{-~\phantom{(}20\phantom{)}} \\ 1\phantom{)} \end{array}$

The quotient is 10 and the remainder is 1.

So, $\frac{21}{2} = 10\frac{1}{2}$.

Therefore, $9\frac{1}{4} + \frac{5}{4} = 10\frac{1}{2}$.

4.55 + 9.73 = 14.28.

To add decimal numbers, we align the decimal points and add the numbers digit by digit from right to left, just like with whole numbers.

We align 4.55 and 9.73 vertically:

$\begin{array}{cc} & 4.55 \\ + & 9.73 \\ \hline & 14.28 \\ \hline \end{array}$

Starting from the rightmost column (hundredths):

$5 + 3 = 8$. Write down 8 in the hundredths place.

Moving to the tenths column:

$5 + 7 = 12$. Write down 2 in the tenths place and carry over 1 to the units place.

Moving to the units column (left of the decimal point):

$4 + 9 + 1$ (carry-over) $= 14$. Write down 14.

The decimal point in the sum is placed directly below the decimal points in the numbers being added.

So, 4.55 + 9.73 = 14.28.

8.76 – 2.68 = 6.08.

To subtract decimal numbers, we align the decimal points and subtract digit by digit from right to left, borrowing when necessary, similar to subtracting whole numbers.

We align 8.76 and 2.68 vertically:

$\begin{array}{cc} & 8.76 \\ - & 2.68 \\ \hline & 6.08 \\ \hline \end{array}$

Starting from the rightmost column (hundredths):

We need to calculate $6 - 8$. Since 6 is smaller than 8, we need to borrow from the tenths place.

Borrow 1 from the 7 in the tenths place. The 7 becomes 6, and the 6 in the hundredths place becomes 16.

Now calculate $16 - 8 = 8$. Write down 8 in the hundredths place.

Moving to the tenths column:

We now have $6 - 6 = 0$. Write down 0 in the tenths place.

Moving to the units column (left of the decimal point):

We calculate $8 - 2 = 6$. Write down 6.

The decimal point in the difference is placed directly below the decimal points in the numbers being subtracted.

So, 8.76 – 2.68 = 6.08.

The value of 50 coins of 50 paisa = Rs 25.

We are given that we have 50 coins, and each coin is worth 50 paisa.

To find the total value, we multiply the number of coins by the value of each coin:

Total value in paisa = Number of coins $\times$ Value per coin

Total value in paisa = $50 \times 50$ paisa

Total value in paisa = 2500 paisa

The question asks for the value in Rupees (Rs). We need to convert the total value from paisa to Rupees.

The relationship between Rupees and paisa is:

To convert paisa to Rupees, we divide the total number of paisa by 100.

Total value in Rupees = $\frac{\text{Total value in paisa}}{100}$

Total value in Rupees = $\frac{2500}{100}$

We can perform the division:

$\frac{2500}{100} = 25$

Total value in Rupees = 25

Alternatively, we can first convert the value of one coin from paisa to Rupees.

Value of one coin = 50 paisa

Value of one coin in Rupees = $\frac{50}{100}$ Rupees = 0.50 Rupees

Now, multiply the number of coins by the value of one coin in Rupees:

Total value in Rupees = Number of coins $\times$ Value per coin in Rupees

Total value in Rupees = $50 \times 0.50$ Rupees

$\begin{array}{cc} & 5 & 0 \\ \times & 0. & 5 & 0 \\ \hline & 0 & 0 & 0 \\ 2 & 5 & 0 & \times \\ + 0 & 0 & \times & \times \\ \hline 2 & 5. & 0 & 0 \\ \hline \end{array}$

Total value in Rupees = 25.00 Rupees

Thus, the value of 50 coins of 50 paisa is Rs 25.

3 Hundredths + 3 tenths = 0.33.

To find the sum, we first need to understand what "Hundredths" and "Tenths" represent in decimal notation.

Tenths are the first decimal place to the right of the decimal point. The value of 3 tenths is $\frac{3}{10}$, which is written as 0.3 in decimal form.

Hundredths are the second decimal place to the right of the decimal point. The value of 3 hundredths is $\frac{3}{100}$, which is written as 0.03 in decimal form.

So, the problem is equivalent to adding the decimal values of 3 tenths and 3 hundredths:

Sum = 0.3 + 0.03

To add decimals, we align the decimal points and add each place value column:

$\begin{array}{cc} & 0.30 \\ + & 0.03 \\ \hline & 0.33 \\ \hline \end{array}$

Adding the hundredths column: $0 + 3 = 3$.

Adding the tenths column: $3 + 0 = 3$.

Adding the units column: $0 + 0 = 0$.

The result is 0.33.

Alternatively, using fractions:

3 Hundredths = $\frac{3}{100}$

3 Tenths = $\frac{3}{10}$

Sum = $\frac{3}{100} + \frac{3}{10}$

To add fractions with different denominators, we find a common denominator. The least common multiple of 100 and 10 is 100.

Convert $\frac{3}{10}$ to an equivalent fraction with a denominator of 100:

$\frac{3}{10} = \frac{3 \times 10}{10 \times 10} = \frac{30}{100}$

Now add the like fractions:

$\frac{3}{100} + \frac{30}{100} = \frac{3 + 30}{100} = \frac{33}{100}$

Convert the resulting fraction to a decimal:

$\frac{33}{100} = 0.33$

Both methods yield the same result, 0.33.

Fractions are called like fractions if they have the same denominator. Fractions with the same numerator but different denominators are called unlike fractions.

Therefore, the statement "Fractions with same numerator are called like fractions" is False.

False

A fraction is in its lowest form (or simplest form) if the only common factor between the numerator and the denominator is $1$. To determine if a fraction is in its lowest form, we can find the greatest common divisor (GCD) of the numerator and the denominator.

The given fraction is $\frac{18}{39}$.

Numerator $= 18$

Denominator $= 39$

Let's find the factors of $18$ and $39$.

Factors of $18$: $1, 2, 3, 6, 9, 18$

Factors of $39$: $1, 3, 13, 39$

The common factors of $18$ and $39$ are $1$ and $3$.

The greatest common divisor (GCD) of $18$ and $39$ is $3$.

Since the GCD of $18$ and $39$ is $3$, which is greater than $1$, the fraction $\frac{18}{39}$ can be simplified by dividing both the numerator and the denominator by their GCD.

Simplifying the fraction:

The simplified fraction is $\frac{6}{13}$. Now, let's check the GCD of $6$ and $13$.

Factors of $6$: $1, 2, 3, 6$

Factors of $13$: $1, 13$

The common factor of $6$ and $13$ is $1$. The GCD is $1$. This confirms that $\frac{6}{13}$ is in its lowest form.

Since the original fraction $\frac{18}{39}$ could be simplified to $\frac{6}{13}$ because their GCD is $3$ (not $1$), it was not in its lowest form.

Therefore, the statement "Fraction $\frac{18}{39}$ is in its lowest form" is false.

True

Two fractions are called equivalent fractions if they represent the same part of a whole, even though their numerators and denominators are different. Equivalent fractions have the same value when simplified to their lowest form.

To check if $\frac{15}{39}$ and $\frac{45}{117}$ are equivalent, we can simplify both fractions to their lowest form or use cross-multiplication.

Method 1: Simplifying to Lowest Form

Simplify the fraction $\frac{15}{39}$:

Find the greatest common divisor (GCD) of $15$ and $39$.

Factors of $15$: $1, 3, 5, 15$

Factors of $39$: $1, 3, 13, 39$

The GCD of $15$ and $39$ is $3$.

Divide the numerator and denominator of $\frac{15}{39}$ by their GCD:

The lowest form of $\frac{15}{39}$ is $\frac{5}{13}$.

Simplify the fraction $\frac{45}{117}$:

Find the greatest common divisor (GCD) of $45$ and $117$.

Factors of $45$: $1, 3, 5, 9, 15, 45$

Factors of $117$: $1, 3, 9, 13, 39, 117$ (Note: $117 = 9 \times 13$)

The GCD of $45$ and $117$ is $9$.

Divide the numerator and denominator of $\frac{45}{117}$ by their GCD:

The lowest form of $\frac{45}{117}$ is $\frac{5}{13}$.

Since both fractions $\frac{15}{39}$ and $\frac{45}{117}$ simplify to the same lowest form $\frac{5}{13}$, they are equivalent fractions.

Method 2: Cross-Multiplication

For two fractions $\frac{a}{b}$ and $\frac{c}{d}$ to be equivalent, the cross-products must be equal, i.e., $a \times d = b \times c$.

For $\frac{15}{39}$ and $\frac{45}{117}$, we check if $15 \times 117 = 39 \times 45$.

Calculate $15 \times 117$:

Calculate $39 \times 45$:

Since $15 \times 117 = 39 \times 45 = 1755$, the cross-products are equal.

Both methods confirm that the fractions are equivalent.

Therefore, the statement "Fractions $\frac{15}{39}$ and $\frac{45}{117}$ are equivalent fractions" is true.

True

A fraction is a number that can be expressed in the form $\frac{a}{b}$, where $a$ is an integer and $b$ is a non-zero integer. This definition is equivalent to the definition of a rational number.

Let the two fractions be $\frac{a}{b}$ and $\frac{c}{d}$, where $a, b, c, d$ are integers, and $b \neq 0$, $d \neq 0$.

To find the sum of these two fractions, we use a common denominator, which is the product of the denominators ($bd$):

Let's examine the numerator and the denominator of the resulting sum:

The numerator is $ad + bc$. Since $a, b, c, d$ are integers, $ad$ is an integer (product of integers), and $bc$ is an integer (product of integers). The sum of two integers ($ad + bc$) is always an integer.

The denominator is $bd$. Since $b$ is a non-zero integer and $d$ is a non-zero integer, their product $bd$ is also always a non-zero integer.

So, the sum $\frac{ad + bc}{bd}$ is in the form $\frac{P}{Q}$, where $P = ad + bc$ is an integer and $Q = bd$ is a non-zero integer.

According to the definition of a fraction, this form represents a fraction.

For example:

Sum of two fractions: $\frac{1}{2} + \frac{1}{3} = \frac{3+2}{6} = \frac{5}{6}$. Here, $5$ is an integer and $6$ is a non-zero integer, so $\frac{5}{6}$ is a fraction.

Sum that results in an integer: $\frac{1}{2} + \frac{1}{2} = \frac{1+1}{2} = \frac{2}{2} = 1$. The integer $1$ can be written as $\frac{1}{1}$, which is a fraction (where $1$ is an integer and $1$ is a non-zero integer).

Thus, the sum of two fractions always results in a number that can be expressed as a ratio of two integers with a non-zero denominator, which is the definition of a fraction (or a rational number).

Therefore, the statement is true.

True

A fraction (or a rational number) is a number that can be expressed in the form $\frac{a}{b}$, where $a$ is an integer and $b$ is a non-zero integer.

Let the two fractions be $\frac{a}{b}$ and $\frac{c}{d}$, where $a, b, c, d$ are integers, and $b \neq 0$, $d \neq 0$.

To subtract the second fraction from the first, we find a common denominator, such as the product of the denominators ($bd$):

Let's examine the numerator and the denominator of the resulting difference:

The numerator is $ad - bc$. Since $a, b, c, d$ are integers, $ad$ is an integer (product of integers), and $bc$ is an integer (product of integers). The difference of two integers ($ad - bc$) is always an integer.

The denominator is $bd$. Since $b$ is a non-zero integer and $d$ is a non-zero integer, their product $bd$ is also always a non-zero integer.

So, the difference $\frac{ad - bc}{bd}$ is in the form $\frac{P}{Q}$, where $P = ad - bc$ is an integer and $Q = bd$ is a non-zero integer.

According to the definition of a fraction, this form represents a fraction.

For example:

Difference of two fractions: $\frac{3}{4} - \frac{1}{5} = \frac{15 - 4}{20} = \frac{11}{20}$. Here, $11$ is an integer and $20$ is a non-zero integer, so $\frac{11}{20}$ is a fraction.

Difference resulting in an integer: $\frac{5}{2} - \frac{1}{2} = \frac{5-1}{2} = \frac{4}{2} = 2$. The integer $2$ can be written as $\frac{2}{1}$, which is a fraction.

Difference resulting in zero: $\frac{1}{3} - \frac{1}{3} = 0$. The integer $0$ can be written as $\frac{0}{1}$, which is a fraction.

Thus, the difference between any two fractions always results in a number that can be expressed as a ratio of two integers with a non-zero denominator, which is the definition of a fraction.

Therefore, the statement is true.

True

The statement describes the fundamental concept of a fraction.

A fraction represents a part of a whole.

When a whole object or quantity is divided into a number of equal parts, each of these parts is a fraction of the whole.

If the whole is divided into $n$ equal parts, then each part represents $\frac{1}{n}$ of the whole. This is known as a unit fraction.

For example:

• If a cake is cut into $6$ equal slices, each slice represents $\frac{1}{6}$ of the cake. Here, the whole (cake) is divided into $6$ equal parts, and each part is the fraction $\frac{1}{6}$.
• If a length of wire is divided into $10$ equal segments, each segment represents $\frac{1}{10}$ of the total length. The whole (wire) is divided into $10$ equal parts, and each part is the fraction $\frac{1}{10}$.

The key condition for a part to represent a fraction of a whole in this way is that the parts must be equal.

Thus, if a whole or an object is divided into a number of equal parts, each such part indeed represents a fraction, specifically a unit fraction where the numerator is $1$ and the denominator is the total number of equal parts.

Therefore, the statement is true.

False

In the place value system, the value a digit represents depends on its position in the number.

Place values decrease by a factor of $10$ as you move from left to right across the decimal point.

The place value of the ones place (the first digit to the left of the decimal point) is $10^0 = 1$.

The place value of the tenths place (the first digit to the right of the decimal point) is $10^{-1} = \frac{1}{10}$ or $0.1$.

Let's compare the place values:

Place value of Ones place $= 1$

Place value of Tenths place $= 0.1$

The relationship between these two place values is:

or equivalently,

The statement claims that the place value of a digit at the tenths place ($0.1$) is $10$ times the same digit at the ones place (interpreting this as the value of that digit in the ones place, $d \times 1$). If we interpret the statement as comparing the place values themselves, it claims the tenths place value ($0.1$) is $10$ times the ones place value ($1$).

This is clearly false.

The correct relationship between adjacent place values is that the place value on the left is $10$ times the place value on the right. So, the Ones place value is $10$ times the Tenths place value.

Therefore, the statement "The place value of a digit at the tenths place is 10 times the same digit at the ones place" is false.

True

In the decimal place value system, as we move from left to right across the decimal point, the place value of each position is $\frac{1}{10}$ times the place value of the position immediately to its left.

Let's consider the place values involved:

• The place value of the tenths place is $10^{-1} = \frac{1}{10} = 0.1$.
• The place value of the hundredths place is $10^{-2} = \frac{1}{100} = 0.01$.

The statement claims that the place value of a digit at the hundredths place is $\frac{1}{10}$ times the same digit at the tenths place. Let the digit be $d$.

The value of the digit $d$ at the hundredths place is $d \times (\text{Hundredths place value}) = d \times 0.01$.

The value of the digit $d$ at the tenths place is $d \times (\text{Tenths place value}) = d \times 0.1$.

The statement says:

Yes, this is true.

Alternatively, if the statement is interpreted as comparing the place values themselves:

Yes, this is also true.

Each place value is $\frac{1}{10}$ of the place value to its left. Therefore, the place value of the hundredths place is $\frac{1}{10}$ times the place value of the tenths place.

Therefore, the statement is true.

False

Rounding a decimal number to a specific number of decimal places means approximating the number to that precision.

To round a decimal to two decimal places, we look at the digit in the third decimal place (the thousandths place).

• If the digit in the third decimal place is $5$ or greater ($\ge 5$), we round up the digit in the second decimal place (the hundredths place).
• If the digit in the third decimal place is less than $5$ ($< 5$), we keep the digit in the second decimal place as it is.

The given decimal number is $3.725$.

• The first decimal place (tenths) has the digit $7$.
• The second decimal place (hundredths) has the digit $2$.
• The third decimal place (thousandths) has the digit $5$.

We need to round to two decimal places. We look at the digit in the third decimal place, which is $5$.

Since the digit in the third decimal place ($5$) is $5$ or greater, we must round up the digit in the second decimal place.

The digit in the second decimal place is $2$. Rounding $2$ up gives $3$.

So, $3.725$ rounded to two decimal places is $3.73$.

The statement claims that $3.725$ is equal to $3.72$ correct to two decimal places. This means it claims the rounded value is $3.72$.

Our calculation shows the correct rounded value is $3.73$, not $3.72$.

Therefore, the statement is false.

To find the decimal form of the fraction $\frac{25}{8}$, we divide 25 by 8.

$\frac{25}{8} = 25 \div 8$

Performing the division:

$25 \div 8 = 3.125$

The decimal form of $\frac{25}{8}$ is indeed 3.125.

Therefore, the statement is True.

We need to check if the decimal 23.2 is equal to the mixed fraction $23\frac{2}{5}$.

Let's convert the decimal 23.2 to a fraction.

$23.2 = 23 + 0.2$

$0.2 = \frac{2}{10}$

$23.2 = 23 + \frac{2}{10}$

We can simplify the fraction $\frac{2}{10}$.

$\frac{\cancel{2}^{1}}{\cancel{10}_{5}} = \frac{1}{5}$

So, $23.2 = 23 + \frac{1}{5} = 23\frac{1}{5}$.

Now let's consider the given mixed fraction $23\frac{2}{5}$.

To compare, we can convert $23\frac{2}{5}$ to a decimal.

$23\frac{2}{5} = 23 + \frac{2}{5}$

To convert $\frac{2}{5}$ to a decimal, we can divide 2 by 5 or make the denominator 10.

$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} = 0.4$

So, $23\frac{2}{5} = 23 + 0.4 = 23.4$.

Comparing the values:

We found that $23.2 = 23\frac{1}{5}$ and $23\frac{2}{5} = 23.4$.

Since $23.2 \neq 23.4$, the given statement is incorrect.

Therefore, the statement "The decimal 23.2 = $23\frac{2}{5}$" is False.

From the given figure, we observe that the figure is divided into a total number of equal parts.

Total number of equal parts = 8

We count the number of shaded parts in the figure.

Number of shaded parts = 3

The fraction represented by the shaded portion is given by:

Fraction = $\frac{\text{Number of shaded parts}}{\text{Total number of equal parts}}$

Fraction = $\frac{3}{8}$

The calculated fraction is $\frac{3}{8}$, which matches the fraction given in the statement.

Therefore, the statement is True.

From the given figure, we count the total number of small equal squares.

Total number of equal parts = 9

Next, we count the number of unshaded small squares.

Number of unshaded parts = 5

The fraction represented by the unshaded portion is given by:

Fraction = $\frac{\text{Number of unshaded parts}}{\text{Total number of equal parts}}$

Fraction = $\frac{5}{9}$

The calculated fraction $\frac{5}{9}$ is the same as the fraction given in the statement.

Therefore, the statement is True.

We are asked to check if the sum of $\frac{25}{19}$ and $\frac{6}{19}$ is equal to $\frac{31}{38}$.

Let's add the fractions on the left side of the equation:

$\frac{25}{19} + \frac{6}{19}$

Since these are like fractions (they have the same denominator, 19), we add the numerators and keep the denominator the same.

$\frac{25}{19} + \frac{6}{19} = \frac{25 + 6}{19}$

$\frac{25 + 6}{19} = \frac{31}{19}$

Now, we compare the result with the right side of the given statement.

The right side is $\frac{31}{38}$.

We found that $\frac{25}{19} + \frac{6}{19} = \frac{31}{19}$.

Is $\frac{31}{19} = \frac{31}{38}$?

No, because the denominators are different (19 vs 38) while the numerators are the same. For two fractions with the same numerator to be equal, their denominators must also be equal.

Therefore, the statement $\frac{25}{19} + \frac{6}{19} = \frac{31}{38}$ is incorrect.

The correct sum is $\frac{31}{19}$.

The statement is False.

We need to verify if the statement $\frac{8}{18} - \frac{8}{15} = \frac{8}{18}$ is true or false by calculating the left side of the equation.

The fractions $\frac{8}{18}$ and $\frac{8}{15}$ are unlike fractions, so we need to find a common denominator to subtract them.

We find the Least Common Multiple (LCM) of the denominators, 18 and 15.

Prime factorization of 18: $2 \times 3^2$

Prime factorization of 15: $3 \times 5$

LCM(18, 15) = $2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 90$.

Now, we convert each fraction to an equivalent fraction with a denominator of 90.

$\frac{8}{18} = \frac{8 \times (90 \div 18)}{18 \times (90 \div 18)} = \frac{8 \times 5}{18 \times 5} = \frac{40}{90}$

$\frac{8}{15} = \frac{8 \times (90 \div 15)}{15 \times (90 \div 15)} = \frac{8 \times 6}{15 \times 6} = \frac{48}{90}$

Now, we perform the subtraction on the left side:

$\frac{8}{18} - \frac{8}{15} = \frac{40}{90} - \frac{48}{90}$

Since the denominators are the same, we subtract the numerators:

$\frac{40 - 48}{90} = \frac{-8}{90}$

The result of the left side is $\frac{-8}{90}$. We can simplify this fraction by dividing the numerator and denominator by their greatest common divisor, which is 2.

$\frac{-8}{90} = \frac{-\cancel{8}^{4}}{\cancel{90}_{45}} = -\frac{4}{45}$

Now, we compare the result $(-\frac{4}{45})$ with the right side of the original statement $(\frac{8}{18})$.

The right side $\frac{8}{18}$ can be simplified by dividing the numerator and denominator by 2.

$\frac{8}{18} = \frac{\cancel{8}^{4}}{\cancel{18}_{9}} = \frac{4}{9}$

We need to check if $-\frac{4}{45} = \frac{4}{9}$.

Clearly, a negative fraction cannot be equal to a positive fraction. Thus, the equality does not hold.

Therefore, the statement "$\frac{8}{18} - \frac{8}{15} = \frac{8}{18}$" is False.

We need to check if the sum of the fractions on the left side of the equation is equal to the fraction on the right side.

Let's calculate the sum $\frac{7}{12} + \frac{11}{12}$.

The fractions have the same denominator (12), so they are like fractions. To add like fractions, we add the numerators and keep the denominator the same.

$\frac{7}{12} + \frac{11}{12} = \frac{7 + 11}{12}$

$\frac{7 + 11}{12} = \frac{18}{12}$

Now, we simplify the resulting fraction $\frac{18}{12}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 6.

$\frac{18}{12} = \frac{\cancel{18}^{3}}{\cancel{12}_{2}} = \frac{3}{2}$

Comparing the simplified result of the left side ($\frac{3}{2}$) with the right side of the original statement ($\frac{3}{2}$), we see that they are equal.

$\frac{3}{2} = \frac{3}{2}$

Therefore, the statement "$\frac{7}{12} + \frac{11}{12} = \frac{3}{2}$" is correct.

The statement is True.

To check if the statement is true, we need to perform the addition 3.03 + 0.016.

When adding decimals, we align the decimal points and add the numbers column by column. We can add trailing zeros to make the number of decimal places equal.

$3.030$

$+ 0.016$

Adding the numbers:

``` 3.030 + 0.016 ------- 3.046 ```

The sum of 3.03 and 0.016 is 3.046.

Now, we compare our calculated sum (3.046) with the result given in the statement (3.019).

$3.046 \neq 3.019$

The values are not equal.

Therefore, the statement "3.03 + 0.016 = 3.019" is incorrect.

The statement is False.

To check if the statement is true, we need to perform the subtraction $42.28 - 3.19$.

When subtracting decimals, we align the decimal points and subtract the numbers column by column, starting from the rightmost digit.

``` 42.28 - 3.19 ------- ```

Subtracting column by column:

In the hundredths place: $8 - 9$. We need to borrow from the tenths place. The 2 in the tenths place becomes 1, and the 8 in the hundredths place becomes 18.

$18 - 9 = 9$.

In the tenths place: $1 - 1 = 0$.

In the ones place: $2 - 3$. We need to borrow from the tens place. The 4 in the tens place becomes 3, and the 2 in the ones place becomes 12.

$12 - 3 = 9$.

In the tens place: $3 - 0 = 3$.

The result of the subtraction is 39.09.

``` 42.28 - 3.19 ------- 39.09 ```

Comparing our calculated result (39.09) with the result given in the statement (39.09), we see that they are equal.

$39.09 = 39.09$

Therefore, the statement "$42.28 – 3.19 = 39.09$" is correct.

The statement is True.

We are asked to determine if the statement $\frac{16}{25} > \frac{13}{25}$ is true or false.

The given fractions are $\frac{16}{25}$ and $\frac{13}{25}$.

We observe that both fractions have the same denominator, which is 25.

When comparing fractions with the same denominator, the fraction with the larger numerator is the greater fraction.

We compare the numerators of the two fractions: 16 and 13.

$16 > 13$

Since the numerator of the first fraction (16) is greater than the numerator of the second fraction (13), and their denominators are the same, the first fraction is greater than the second fraction.

Thus, $\frac{16}{25} > \frac{13}{25}$.

The inequality in the statement matches our comparison.

Therefore, the statement "$\frac{16}{25} > \frac{13}{25}$" is correct.

The statement is True.

We are asked to determine if the statement $19.25 < 19.053$ is true or false.

To compare two decimal numbers, we compare the digits starting from the leftmost digit (highest place value) and move to the right.

The two numbers are $19.25$ and $19.053$.

We compare the digits:

1. Compare the tens digits: The tens digit in both numbers is $1$. They are equal.

2. Compare the ones digits: The ones digit in both numbers is $9$. They are equal.

3. Compare the tenths digits: The tenths digit in $19.25$ is $2$. The tenths digit in $19.053$ is $0$.

Comparing the tenths digits, we have $2 > 0$.

Since the tenths digit of $19.25$ ($2$) is greater than the tenths digit of $19.053$ ($0$), the number $19.25$ is greater than $19.053$.

So, $19.25 > 19.053$.

The given statement claims that $19.25 < 19.053$. This contradicts our finding that $19.25 > 19.053$.

Therefore, the statement is incorrect.

The statement is False.

We are asked to determine if the statement $13.730 = 13.73$ is true or false.

The number $13.730$ means 13 units, 7 tenths, 3 hundredths, and 0 thousandths.

The number $13.73$ means 13 units, 7 tenths, and 3 hundredths.

In decimal numbers, adding or removing zeros at the end of the decimal part does not change the value of the number.

For example, $0.5 = 0.50 = 0.500$, and so on.

Applying this rule, the number $13.730$ has a zero at the end of the decimal part. This zero is in the thousandths place but does not contribute to the value beyond the hundredths place because it is a trailing zero.

So, $13.730$ is equivalent to $13.73$.

$13.730 = 13.73$

The equality in the statement matches our understanding of decimal numbers.

Therefore, the statement "13.730 = 13.73" is correct.

The statement is True.

To compare the fractions $\frac{11}{16}$ and $\frac{14}{15}$, we can use the cross-multiplication method.

We multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction.

Compare $11 \times 15$ and $16 \times 14$.

$11 \times 15 = 165$

$16 \times 14 = 224$

Now, we compare the results of the cross-multiplication:

$165 \_ 224$

Since $165$ is less than $224$, we have:

$165 < 224$

This inequality holds for the fractions as well:

$\frac{11}{16} < \frac{14}{15}$

Therefore, the correct symbol to fill in the blank is '$<$'.

$\frac{11}{16}$ < $\frac{14}{15}$

We need to compare the fractions $\frac{8}{15}$ and $\frac{95}{14}$.

We can observe the nature of these fractions.

The first fraction $\frac{8}{15}$ has a numerator (8) that is smaller than its denominator (15). This means the value of the fraction is less than 1.

$8 < 15 \implies \frac{8}{15} < 1$

The second fraction $\frac{95}{14}$ has a numerator (95) that is larger than its denominator (14). This means the value of the fraction is greater than 1. It is an improper fraction.

$95 > 14 \implies \frac{95}{14} > 1$

Since $\frac{8}{15}$ is less than 1 and $\frac{95}{14}$ is greater than 1, we can conclude that $\frac{8}{15}$ is less than $\frac{95}{14}$.

$\frac{8}{15} < \frac{95}{14}$

Alternatively, we can use the cross-multiplication method.

Compare $8 \times 14$ and $15 \times 95$.

$8 \times 14 = 112$

$15 \times 95$

$\begin{array}{cc}& & & 9 & 5 \\ \times & & & 1 & 5 \\ \hline && 4 & 7 & 5 \\ & 9 & 5 & \times \\ \hline 1 & 4 & 2 & 5 \\ \hline \end{array}$

$15 \times 95 = 1425$

Compare the results: $112$ and $1425$.

$112 < 1425$

This confirms that:

$\frac{8}{15} < \frac{95}{14}$

Therefore, the correct symbol to fill in the blank is '$<$'.

$\frac{8}{15}$ < $\frac{95}{14}$

To compare the fractions $\frac{12}{75}$ and $\frac{32}{200}$, we can simplify both fractions to their lowest terms.

Consider the first fraction $\frac{12}{75}$.

We find the greatest common divisor (GCD) of 12 and 75.

Factors of 12 are 1, 2, 3, 4, 6, 12.

Factors of 75 are 1, 3, 5, 15, 25, 75.

The GCD of 12 and 75 is 3.

We divide both the numerator and the denominator by 3:

$\frac{12}{75} = \frac{12 \div 3}{75 \div 3} = \frac{4}{25}$

Now consider the second fraction $\frac{32}{200}$.

We find the greatest common divisor (GCD) of 32 and 200.

Factors of 32 are 1, 2, 4, 8, 16, 32.

Factors of 200 are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200.

The GCD of 32 and 200 is 8.

We divide both the numerator and the denominator by 8:

$\frac{32}{200} = \frac{32 \div 8}{200 \div 8} = \frac{4}{25}$

We have simplified both fractions:

$\frac{12}{75}$ simplifies to $\frac{4}{25}$.

$\frac{32}{200}$ simplifies to $\frac{4}{25}$.

Since the simplified forms of the two fractions are equal, the original fractions are also equal.

$\frac{12}{75} = \frac{32}{200}$

Therefore, the correct symbol to fill in the blank is '='.

$\frac{12}{75}$ = $\frac{32}{200}$

To compare the decimal numbers 3.25 and 3.4, we compare the digits from left to right, starting with the largest place value.

The numbers are 3.25 and 3.4. We can write 3.4 as 3.40 to have the same number of decimal places for easier comparison.

Numbers: 3.25 and 3.40

1. Compare the whole number parts: The whole number part is 3 for both numbers. They are equal.

2. Compare the digits in the tenths place: The digit in the tenths place for 3.25 is 2. The digit in the tenths place for 3.40 is 4.

Comparing these digits, we see that $2 < 4$.

Since the digit in the tenths place of 3.25 (which is 2) is less than the digit in the tenths place of 3.4 (which is 4), the number 3.25 is less than 3.4.

$3.25 < 3.4$

Therefore, the correct symbol to fill in the blank is '$<$'.

3.25 < 3.4

To compare the fraction $\frac{18}{25}$ and the decimal $1.3$, we can convert the fraction to a decimal.

To convert $\frac{18}{25}$ to a decimal, we can divide the numerator (18) by the denominator (25).

Alternatively, we can make the denominator 100 by multiplying both the numerator and the denominator by 4.

$\frac{18}{25} = \frac{18 \times 4}{25 \times 4} = \frac{72}{100}$

Converting the fraction $\frac{72}{100}$ to a decimal:

$\frac{72}{100} = 0.72$

Now we compare the decimal value of the fraction (0.72) with the given decimal (1.3).

We are comparing $0.72$ and $1.3$.

Compare the whole number parts: $0 < 1$.

Since the whole number part of $0.72$ is less than the whole number part of $1.3$, the number $0.72$ is less than $1.3$.

$0.72 < 1.3$

This means that the original fraction is also less than the decimal.

$\frac{18}{25} < 1.3$

Therefore, the correct symbol to fill in the blank is '$<$'.

$\frac{18}{25}$ < $1.3$

To compare the decimal $6.25$ and the fraction $\frac{25}{4}$, we can convert the fraction to a decimal.

To convert $\frac{25}{4}$ to a decimal, we divide the numerator (25) by the denominator (4).

$25 \div 4$

``` ``` $\begin{array}{r} 6.25 \\ 4{\overline{\smash{\big)}\,25.00}} \\ \underline{-24}\phantom{.00} \\ 10\phantom{0} \\ \underline{-~8}\phantom{0} \\ 20 \\ \underline{-20} \\ 0 \end{array}$

So, the decimal form of $\frac{25}{4}$ is $6.25$.

Now we compare the given decimal (6.25) with the decimal value of the fraction (6.25).

We are comparing $6.25$ and $6.25$.

The two values are equal.

$6.25 = 6.25$

This means that the original decimal and the fraction are equal.

$6.25 = \frac{25}{4}$

Therefore, the correct symbol to fill in the blank is '='.

$6.25$ = $\frac{25}{4}$

From the given figure, we can observe that the circle is divided into equal parts.

Let's count the total number of equal parts in the figure.

Total number of equal parts = 8

Now, let's count the number of shaded parts in the figure.

Number of shaded parts = 3

The fraction represented by the shaded portion is given by the formula:

Fraction = $\frac{\text{Number of shaded parts}}{\text{Total number of equal parts}}$

Substituting the values, we get:

Fraction = $\frac{3}{8}$

Thus, the fraction represented by the shaded portion of the adjoining figure is $\frac{3}{8}$.

The answer is $\frac{3}{8}$.

From the given figure, we first count the total number of equal parts into which the figure is divided. The figure is a rectangle divided into small equal squares.

Total number of equal squares = 12

Next, we count the number of unshaded squares in the figure.

Number of unshaded squares = 5

The fraction represented by the unshaded portion is given by the ratio of the number of unshaded parts to the total number of equal parts.

Fraction of unshaded portion = $\frac{\text{Number of unshaded squares}}{\text{Total number of equal squares}}$

Fraction of unshaded portion = $\frac{5}{12}$

Thus, the fraction represented by the unshaded portion of the adjoining figure is $\frac{5}{12}$.

The answer is $\frac{5}{12}$.

Given:

Total number of fruit cakes = 1

Number of persons among whom the cake is divided = 6

To Find:

The part of the cake each person received.

Solution:

Since Ali divided the fruit cake equally among six persons, each person will receive a fraction of the cake.

The part of the cake each person received is calculated by dividing the total number of cakes by the number of persons.

Part of cake per person = $\frac{\text{Total number of cakes}}{\text{Number of persons}}$

Part of cake per person = $\frac{1}{6}$

Thus, each person received $\frac{1}{6}$ part of the fruit cake.

Final Answer:

Each person received $\frac{1}{6}$ part of the fruit cake.

Given:

The numbers to be arranged are: 12.142, 12.124, 12.104, 12.401, and 12.214.

To Arrange:

Arrange the given decimal numbers in ascending order (from smallest to largest).

Solution:

To arrange decimal numbers in ascending order, we compare them digit by digit from left to right, starting with the leftmost digit.

All the given numbers have the same whole number part, which is 12.

Now, we compare the digits in the tenths place:

12.142, 12.124, 12.104, 12.401, 12.214

The tenths digits are 1, 1, 1, 4, and 2.

The smallest tenths digit is 1. So, we compare the numbers with 1 in the tenths place: 12.142, 12.124, 12.104.

Now, we compare the digits in the hundredths place for these three numbers:

12.142, 12.124, 12.104

The hundredths digits are 4, 2, and 0.

The smallest hundredths digit is 0. So, 12.104 is the smallest number among these.

Next smallest is 2. So, 12.124 is the next smallest.

The largest among these three is 12.142 (hundredths digit is 4).

So far, the order is: 12.104, 12.124, 12.142.

Now, we consider the remaining numbers: 12.401 and 12.214. Their tenths digits are 4 and 2 respectively.

Comparing the tenths digits, 2 is smaller than 4. So, 12.214 is smaller than 12.401.

The largest number among all is 12.401.

Combining all the numbers in ascending order, we get:

12.104, 12.124, 12.142, 12.214, 12.401.

Final Answer:

The given numbers arranged in ascending order are:

12.104, 12.124, 12.142, 12.214, 12.401.

Given:

Digits available are 1, 5, 3, and 8.

We need to form a four-digit decimal number less than 1 using these digits exactly once.

To Find:

The largest four-digit decimal number less than 1 using the given digits.

Solution:

A decimal number less than 1 has a whole number part of 0. The number will be in the form $0.abcd$, where $a, b, c, d$ are the digits 1, 5, 3, and 8 in some order.

To make the decimal number as large as possible, we need to place the largest digits in the positions with the highest place value after the decimal point (tenths, hundredths, thousandths, etc.).

The available digits are 1, 3, 5, and 8.

Arranging these digits in descending order gives us 8, 5, 3, 1.

To form the largest possible decimal number less than 1, we place these digits in order after the decimal point.

The largest digit, 8, goes in the tenths place.

The next largest digit, 5, goes in the hundredths place.

The next largest digit, 3, goes in the thousandths place.

The smallest digit, 1, goes in the ten-thousandths place.

So, the number is 0.8531.

Final Answer:

The largest four-digit decimal number less than 1 using the digits 1, 5, 3 and 8 once is 0.8531.

Given:

The digits available are 2, 4, 5, and 3.

We need to form the smallest four-digit decimal number using these digits exactly once.

To Find:

The smallest four-digit decimal number using the digits 2, 4, 5 and 3 once.

Solution:

To form the smallest possible number using a given set of digits, we should arrange the digits in ascending order and place the digit with the smallest value in the highest place value position (the leftmost position).

The given digits are 2, 4, 5, and 3.

Arranging these digits in ascending order gives us 2, 3, 4, 5.

A "four-digit decimal number" using these digits, especially in the context of similar problems, often implies a number with a decimal point where these four digits are used to form the number.

To create the smallest possible value, the smallest digit should be in the position with the largest place value.

If we interpret "four digit decimal number" as a number with four digits after the decimal point (preceded by 0, as done in the previous question), we arrange the digits 2, 3, 4, 5 in ascending order after the decimal point.

The smallest arrangement of the digits 2, 3, 4, 5 is 2345.

Placing this after the decimal point with 0 in the whole number part gives 0.2345.

This number uses the digits 2, 3, 4, and 5 exactly once, and it is a decimal number.

Final Answer:

The smallest four-digit decimal number using the digits 2, 4, 5 and 3 once is 0.2345.

Given:

The fraction is $\frac{11}{20}$.

To Express:

Express the given fraction as a decimal number.

Solution:

To express a fraction as a decimal, we divide the numerator by the denominator.

Here, the numerator is 11 and the denominator is 20.

We need to calculate $11 \div 20$.

$\frac{11}{20} = 11 \div 20$

Performing the division:

$11 \div 20 = 0.55$

Alternate Method (Converting denominator to a power of 10):

We can convert the denominator 20 to a power of 10 by multiplying both the numerator and the denominator by 5.

$\frac{11}{20} = \frac{11 \times 5}{20 \times 5} = \frac{55}{100}$

Now, expressing $\frac{55}{100}$ as a decimal is straightforward as the denominator is 100.

$\frac{55}{100} = 0.55$

Final Answer:

The decimal form of $\frac{11}{20}$ is 0.55.

Given:

The mixed fraction is $6\frac{2}{3}$.

To Express:

Express the given mixed fraction as an improper fraction.

Solution:

A mixed fraction consists of a whole number part and a fractional part. To convert a mixed fraction $a\frac{b}{c}$ to an improper fraction, we use the formula:

Improper fraction = $\frac{(\text{Whole number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}$

In the given mixed fraction $6\frac{2}{3}$, the whole number is 6, the numerator of the fractional part is 2, and the denominator is 3.

Applying the formula:

$6\frac{2}{3} = \frac{(6 \times 3) + 2}{3}$

$6\frac{2}{3} = \frac{18 + 2}{3}$

$6\frac{2}{3} = \frac{20}{3}$

Thus, the improper fraction is $\frac{20}{3}$.

Final Answer:

The improper fraction form of $6\frac{2}{3}$ is $\frac{20}{3}$.

Given:

The mixed fraction is $3\frac{2}{5}$.

To Express:

Express the given mixed fraction as a decimal number.

Solution:

A mixed fraction consists of a whole number part and a fractional part.

In the mixed fraction $3\frac{2}{5}$, the whole number part is 3 and the fractional part is $\frac{2}{5}$.

To express the mixed fraction as a decimal, we can convert the fractional part to a decimal and add it to the whole number part.

The fractional part is $\frac{2}{5}$. To convert this to a decimal, we can divide the numerator (2) by the denominator (5).

$\frac{2}{5} = 2 \div 5 = 0.4$

Now, add the whole number part to this decimal:

$3 + 0.4 = 3.4$

Alternate Method:

Convert the mixed fraction to an improper fraction first.

$3\frac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$

Now, divide the numerator (17) by the denominator (5) to get the decimal form.

$\frac{17}{5} = 17 \div 5 = 3.4$

Final Answer:

The decimal form of $3\frac{2}{5}$ is 3.4.

Given:

The decimal number is 0.041.

To Express:

Express the given decimal number as a fraction.

Solution:

To express a decimal as a fraction, we look at the place value of the last digit.

In the number 0.041:

The digit 0 is in the tenths place.

The digit 4 is in the hundredths place.

The digit 1 is in the thousandths place.

Since the last digit (1) is in the thousandths place, the denominator of the fraction will be 1000.

The numerator will be the number without the decimal point (considering only the digits after the decimal point), which is 041, or simply 41.

So, 0.041 can be written as $\frac{41}{1000}$.

We need to check if the fraction $\frac{41}{1000}$ can be simplified. We find the greatest common divisor (GCD) of 41 and 1000.

41 is a prime number.

We check if 1000 is divisible by 41. $1000 \div 41$ is not a whole number.

Therefore, 41 and 1000 have no common factors other than 1, which means the fraction is already in its simplest form.

Final Answer:

The fraction form of 0.041 is $\frac{41}{1000}$.

Given:

The decimal number is 6.03.

To Express:

Express the given decimal number as a mixed fraction.

Solution:

A decimal number can be separated into its whole number part and its decimal part.

In the number 6.03, the whole number part is 6.

The decimal part is 0.03.

To express the decimal part 0.03 as a fraction, we look at the place value of the last digit.

The digit 3 is in the hundredths place (two decimal places).

So, 0.03 can be written as $\frac{3}{100}$.

The number 6.03 is the sum of the whole number part and the fractional part:

$6.03 = 6 + 0.03 = 6 + \frac{3}{100}$

Combining the whole number and the fraction gives the mixed fraction.

$6\frac{3}{100}$

We check if the fractional part $\frac{3}{100}$ can be simplified. The numerator is 3 and the denominator is 100. The only positive divisors of 3 are 1 and 3. The number 100 is not divisible by 3. Therefore, the fraction $\frac{3}{100}$ is in its simplest form.

Final Answer:

The mixed fraction form of 6.03 is $6\frac{3}{100}$.

Given:

The mass is 5201 grams (g).

To Convert:

Convert 5201 g to kilograms (kg).

Solution:

We know that 1 kilogram (kg) is equal to 1000 grams (g).

$1 \text{ kg} = 1000 \text{ g}$

To convert grams to kilograms, we need to divide the number of grams by 1000.

Conversion factor: $1 \text{ g} = \frac{1}{1000} \text{ kg}$

So, to convert 5201 g to kg, we multiply 5201 by the conversion factor $\frac{1}{1000}$.

$5201 \text{ g} = 5201 \times \frac{1}{1000} \text{ kg}$

$5201 \text{ g} = \frac{5201}{1000} \text{ kg}$

Dividing 5201 by 1000:

$\frac{5201}{1000} = 5.201$

Therefore, 5201 g is equal to 5.201 kg.

Final Answer:

5201 g is equal to 5.201 kg.

Given:

The amount is 2009 paise.

To Convert:

Convert 2009 paise to rupees and express the result as a mixed fraction.

Solution:

We know the relationship between rupees and paise:

$1 \textsf{₹} = 100 \text{ paise}$

To convert paise to rupees, we divide the amount in paise by 100.

$2009 \text{ paise} = \frac{2009}{100} \textsf{₹}$

To express $\frac{2009}{100}$ as a mixed fraction, we divide 2009 by 100.

$2009 \div 100$

We get a quotient of 20 and a remainder of 9.

So, $\frac{2009}{100}$ can be written as the mixed fraction $20\frac{9}{100}$.

The whole number part is the quotient, and the numerator of the fractional part is the remainder. The denominator remains the same as the original fraction's denominator.

$20\frac{9}{100}$

We check if the fractional part $\frac{9}{100}$ can be simplified. The factors of 9 are 1, 3, and 9. The factors of 100 include 1, 2, 4, 5, 10, 20, 25, 50, 100. The only common factor is 1. So, the fraction $\frac{9}{100}$ is already in its simplest form.

Final Answer:

2009 paise is equal to $20\frac{9}{100}$ rupees.

Given:

The length is 1537 centimeters (cm).

To Convert:

Convert 1537 cm to meters (m) and express the result as an improper fraction.

Solution:

We know the relationship between meters and centimeters:

$1 \text{ m} = 100 \text{ cm}$

To convert centimeters to meters, we need to divide the number of centimeters by 100.

Conversion factor: $1 \text{ cm} = \frac{1}{100} \text{ m}$

So, to convert 1537 cm to m, we multiply 1537 by the conversion factor $\frac{1}{100}$.

$1537 \text{ cm} = 1537 \times \frac{1}{100} \text{ m}$

$1537 \text{ cm} = \frac{1537}{100} \text{ m}$

The result is already in the form of an improper fraction $\frac{1537}{100}$ because the numerator (1537) is greater than the denominator (100).

We should check if the fraction can be simplified. The prime factors of 100 are $2^2 \times 5^2$. We check if 1537 is divisible by 2 or 5. 1537 is not divisible by 2 (it is odd). 1537 does not end in 0 or 5, so it is not divisible by 5. We can try other prime factors, but it is likely that the fraction is in its simplest form for this type of problem.

Since the question specifically asks for the result as an improper fraction, $\frac{1537}{100}$ is the required answer.

Final Answer:

1537 cm is equal to $\frac{1537}{100}$ m as an improper fraction.

Given:

The length is 2435 meters (m).

To Convert:

Convert 2435 m to kilometers (km) and express the result as a mixed fraction.

Solution:

We know the relationship between kilometers and meters:

$1 \text{ km} = 1000 \text{ m}$

To convert meters to kilometers, we divide the number of meters by 1000.

Conversion factor: $1 \text{ m} = \frac{1}{1000} \text{ km}$

So, to convert 2435 m to km, we divide 2435 by 1000.

$2435 \text{ m} = \frac{2435}{1000} \text{ km}$

This is an improper fraction. To express it as a mixed fraction, we divide the numerator by the denominator.

$2435 \div 1000$

Performing the division, we find that 1000 goes into 2435 two times with a remainder.

$2435 = 2 \times 1000 + 435$

The quotient is 2, which is the whole number part of the mixed fraction.

The remainder is 435, which is the numerator of the fractional part.

The denominator remains 1000.

So, the mixed fraction is $2\frac{435}{1000}$.

Now, we simplify the fractional part $\frac{435}{1000}$. Both numbers are divisible by 5.

$\frac{435}{1000} = \frac{435 \div 5}{1000 \div 5} = \frac{87}{200}$

The fraction $\frac{87}{200}$ cannot be simplified further as the greatest common divisor of 87 and 200 is 1.

Therefore, the mixed fraction in simplest form is $2\frac{87}{200}$.

Final Answer:

2435 m is equal to $2\frac{87}{200}$ km.

Given:

The fractions to be arranged are $\frac{2}{3}$, $\frac{3}{4}$, $\frac{1}{2}$, and $\frac{5}{6}$.

To Arrange:

Arrange the given fractions in ascending order (from smallest to largest).

Solution:

To compare and arrange fractions, we need to express them with a common denominator. The denominators are 3, 4, 2, and 6.

We find the least common multiple (LCM) of the denominators.

LCM of 3, 4, 2, 6:

LCM = $2 \times 2 \times 3 = 12$.

Now, we convert each fraction into an equivalent fraction with a denominator of 12.

For $\frac{2}{3}$: Multiply numerator and denominator by $\frac{12}{3} = 4$.

$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

For $\frac{3}{4}$: Multiply numerator and denominator by $\frac{12}{4} = 3$.

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

For $\frac{1}{2}$: Multiply numerator and denominator by $\frac{12}{2} = 6$.

$\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}$

For $\frac{5}{6}$: Multiply numerator and denominator by $\frac{12}{6} = 2$.

$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

The fractions with the common denominator are $\frac{8}{12}$, $\frac{9}{12}$, $\frac{6}{12}$, and $\frac{10}{12}$.

To arrange these fractions in ascending order, we arrange their numerators in ascending order:

6, 8, 9, 10.

So, the equivalent fractions in ascending order are:

$\frac{6}{12}$, $\frac{8}{12}$, $\frac{9}{12}$, $\frac{10}{12}$.

Replacing these with the original fractions, we get:

$\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{5}{6}$.

Final Answer:

The fractions in ascending order are:

$\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{5}{6}$.

Given:

The fractions to be arranged are $\frac{6}{7}$, $\frac{7}{8}$, $\frac{4}{5}$, and $\frac{3}{4}$.

To Arrange:

Arrange the given fractions in descending order (from largest to smallest).

Solution:

To compare and arrange fractions, we need to express them with a common denominator. The denominators are 7, 8, 5, and 4.

We find the least common multiple (LCM) of the denominators.

LCM of 7, 8, 5, 4:

The prime factors are:

7 = 7

8 = $2 \times 2 \times 2 = 2^3$

5 = 5

4 = $2 \times 2 = 2^2$

LCM = $2^3 \times 5 \times 7 = 8 \times 5 \times 7 = 280$.

Now, we convert each fraction into an equivalent fraction with a denominator of 280.

For $\frac{6}{7}$: Multiply numerator and denominator by $\frac{280}{7} = 40$.

$\frac{6}{7} = \frac{6 \times 40}{7 \times 40} = \frac{240}{280}$

For $\frac{7}{8}$: Multiply numerator and denominator by $\frac{280}{8} = 35$.

$\frac{7}{8} = \frac{7 \times 35}{8 \times 35} = \frac{245}{280}$

For $\frac{4}{5}$: Multiply numerator and denominator by $\frac{280}{5} = 56$.

$\frac{4}{5} = \frac{4 \times 56}{5 \times 56} = \frac{224}{280}$

For $\frac{3}{4}$: Multiply numerator and denominator by $\frac{280}{4} = 70$.

$\frac{3}{4} = \frac{3 \times 70}{4 \times 70} = \frac{210}{280}$

The fractions with the common denominator are $\frac{240}{280}$, $\frac{245}{280}$, $\frac{224}{280}$, and $\frac{210}{280}$.

To arrange these fractions in descending order, we arrange their numerators in descending order:

245, 240, 224, 210.

So, the equivalent fractions in descending order are:

$\frac{245}{280}$, $\frac{240}{280}$, $\frac{224}{280}$, $\frac{210}{280}$.

Replacing these with the original fractions, we get:

$\frac{7}{8}$, $\frac{6}{7}$, $\frac{4}{5}$, $\frac{3}{4}$.

Final Answer:

The fractions in descending order are:

$\frac{7}{8}$, $\frac{6}{7}$, $\frac{4}{5}$, $\frac{3}{4}$.

Given:

The fraction is $\frac{3}{4}$.

The target denominator is 44.

To Find:

Express $\frac{3}{4}$ as an equivalent fraction with a denominator of 44.

Solution:

To write an equivalent fraction with a different denominator, we need to multiply both the numerator and the denominator by the same number.

We want to change the denominator from 4 to 44.

Let the new fraction be $\frac{x}{44}$. We have:

$\frac{3}{4} = \frac{x}{44}$

To find the value of $x$, we first determine what number the original denominator (4) was multiplied by to get the new denominator (44).

Multiplier $= \frac{\text{New Denominator}}{\text{Original Denominator}} = \frac{44}{4} = 11$

So, the denominator was multiplied by 11.

To keep the fraction equivalent, we must also multiply the numerator (3) by the same number (11).

New Numerator $= \text{Original Numerator} \times \text{Multiplier} = 3 \times 11 = 33$

Thus, the equivalent fraction with a denominator of 44 is $\frac{33}{44}$.

Final Answer:

The fraction $\frac{3}{4}$ written as a fraction with denominator 44 is $\frac{33}{44}$.

Given:

The fraction is $\frac{5}{6}$.

The target numerator is 60.

To Find:

Express $\frac{5}{6}$ as an equivalent fraction with a numerator of 60.

Solution:

To write an equivalent fraction with a different numerator, we need to multiply both the numerator and the denominator by the same number.

We want to change the numerator from 5 to 60.

Let the new fraction be $\frac{60}{y}$. We have:

$\frac{5}{6} = \frac{60}{y}$

To find the value of $y$, we first determine what number the original numerator (5) was multiplied by to get the new numerator (60).

Multiplier $= \frac{\text{New Numerator}}{\text{Original Numerator}} = \frac{60}{5} = 12$

So, the numerator was multiplied by 12.

To keep the fraction equivalent, we must also multiply the denominator (6) by the same number (12).

New Denominator $= \text{Original Denominator} \times \text{Multiplier} = 6 \times 12 = 72$

Thus, the equivalent fraction with a numerator of 60 is $\frac{60}{72}$.

Final Answer:

The fraction $\frac{5}{6}$ written as a fraction with numerator 60 is $\frac{60}{72}$.

Given:

The improper fraction is $\frac{129}{8}$.

To Express:

Express the given improper fraction as a mixed fraction.

Solution:

To convert an improper fraction to a mixed fraction, we divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the numerator of the fractional part, and the denominator remains the same.

We divide 129 by 8.

$129 \div 8$

Performing the division:

129 divided by 8. $12 \div 8 = 1$ with remainder 4.

Bring down 9, we have 49.

$49 \div 8 = 6$ with remainder 1.

So, the quotient is 16 and the remainder is 1.

Whole number part = Quotient = 16

Numerator of fractional part = Remainder = 1

Denominator of fractional part = Original Denominator = 8

The mixed fraction is $16\frac{1}{8}$.

We check if the fractional part $\frac{1}{8}$ can be simplified. The numerator is 1, so the fraction is in its simplest form.

Final Answer:

The mixed fraction form of $\frac{129}{8}$ is $16\frac{1}{8}$.

Given:

The decimal number is 20.83.

To Round Off:

Round off 20.83 to the nearest tenths.

Solution:

To round off a decimal number to the nearest tenths, we look at the digit in the hundredths place.

In the number 20.83, the digit in the tenths place is 8.

The digit in the hundredths place (the digit to the right of the tenths place) is 3.

The rule for rounding is:

If the digit in the hundredths place is 5 or greater, we round up the digit in the tenths place.

If the digit in the hundredths place is less than 5, we keep the digit in the tenths place as it is.

In this case, the digit in the hundredths place is 3, which is less than 5.

So, we keep the digit in the tenths place (8) as it is and drop all the digits to its right.

The rounded number is 20.8.

Final Answer:

20.83 rounded off to the nearest tenths is 20.8.

Given:

The decimal number is 75.195.

To Round Off:

Round off 75.195 to the nearest hundredths.

Solution:

To round off a decimal number to the nearest hundredths, we look at the digit in the thousandths place.

In the number 75.195, the digit in the hundredths place is 9.

The digit in the thousandths place (the digit to the right of the hundredths place) is 5.

The rule for rounding is:

If the digit in the thousandths place is 5 or greater, we round up the digit in the hundredths place.

If the digit in the thousandths place is less than 5, we keep the digit in the hundredths place as it is.

In this case, the digit in the thousandths place is 5, which is 5 or greater.

So, we round up the digit in the hundredths place (9).

Rounding up 9 gives 10. We write 0 in the hundredths place and carry over 1 to the tenths place.

The digit in the tenths place is 1. Adding the carried-over 1, the new digit in the tenths place is $1 + 1 = 2$.

All digits to the right of the hundredths place are dropped.

The rounded number is 75.20.

It is important to keep the zero in the hundredths place (75.20) to indicate that the rounding was done to the hundredths place.

Final Answer:

75.195 rounded off to the nearest hundredths is 75.20.

Given:

The decimal number is 27.981.

To Round Off:

Round off 27.981 to the nearest tenths.

Solution:

To round off a decimal number to the nearest tenths, we need to look at the digit in the hundredths place.

In the number 27.981:

The digit in the tenths place is 9.

The digit in the hundredths place (the digit to the right of the tenths place) is 8.

We apply the rounding rule based on the digit in the hundredths place:

If the digit is 5 or greater, we increase the tenths digit by 1.

If the digit is less than 5, we keep the tenths digit as it is.

In this case, the hundredths digit is 8, which is 5 or greater.

Therefore, we increase the tenths digit (9) by 1.

Rounding up 9 results in 10. We write 0 in the tenths place and carry over 1 to the units place.

The units digit is 7. Adding the carried-over 1, the units digit becomes $7 + 1 = 8$.

The digits to the right of the tenths place are dropped.

The rounded number is 28.0.

Note that we write 28.0 to show that the number has been rounded to the tenths place.

Final Answer:

27.981 rounded off to the nearest tenths is 28.0.

Given:

The fractions to be added are $\frac{3}{8}$ and $\frac{2}{3}$.

To Find:

The sum of the given fractions.

Solution:

To add fractions with different denominators, we first find a common denominator, which is the least common multiple (LCM) of the denominators.

The denominators are 8 and 3.

The LCM of 8 and 3 is 24.

Now, we convert each fraction to an equivalent fraction with a denominator of 24.

For $\frac{3}{8}$: Multiply the numerator and denominator by $\frac{24}{8} = 3$.

$\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$

For $\frac{2}{3}$: Multiply the numerator and denominator by $\frac{24}{3} = 8$.

$\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}$

Now that the fractions have the same denominator, we can add them by adding their numerators and keeping the common denominator.

$\frac{3}{8} + \frac{2}{3} = \frac{9}{24} + \frac{16}{24}$

Sum $= \frac{9 + 16}{24} = \frac{25}{24}$

The result is an improper fraction $\frac{25}{24}$. We can also express this as a mixed fraction.

Divide 25 by 24: $25 \div 24 = 1$ with a remainder of $25 - (1 \times 24) = 1$.

So, $\frac{25}{24} = 1\frac{1}{24}$.

Final Answer:

The sum of the fractions $\frac{3}{8}$ and $\frac{2}{3}$ is $\frac{25}{24}$ or $1\frac{1}{24}$.

Given:

The numbers to be added are the fraction $\frac{3}{8}$ and the mixed fraction $6\frac{3}{4}$.

To Find:

The sum of $\frac{3}{8}$ and $6\frac{3}{4}$.

Solution:

To add a fraction and a mixed fraction, we can first convert the mixed fraction into an improper fraction. Alternatively, we can add the whole number part separately and then add the fractional parts.

Method 1: Convert mixed fraction to improper fraction.

The mixed fraction is $6\frac{3}{4}$.

To convert $6\frac{3}{4}$ to an improper fraction, we multiply the whole number (6) by the denominator (4) and add the numerator (3). The denominator remains the same.

$6\frac{3}{4} = \frac{(6 \times 4) + 3}{4} = \frac{24 + 3}{4} = \frac{27}{4}$

Now we need to add $\frac{3}{8}$ and $\frac{27}{4}$.

To add fractions with different denominators, we find a common denominator. The denominators are 8 and 4.

The least common multiple (LCM) of 8 and 4 is 8.

Convert $\frac{27}{4}$ to an equivalent fraction with a denominator of 8. We multiply the numerator and denominator by $\frac{8}{4} = 2$.

$\frac{27}{4} = \frac{27 \times 2}{4 \times 2} = \frac{54}{8}$

Now add the equivalent fractions:

$\frac{3}{8} + \frac{27}{4} = \frac{3}{8} + \frac{54}{8}$

Add the numerators and keep the common denominator:

$\frac{3 + 54}{8} = \frac{57}{8}$

The result is $\frac{57}{8}$. This is an improper fraction. We can convert it back to a mixed fraction by dividing the numerator (57) by the denominator (8).

$57 \div 8 = 7$ with a remainder of $57 - (8 \times 7) = 57 - 56 = 1$.

So, $\frac{57}{8} = 7\frac{1}{8}$.

Alternate Method:

Add the whole number part and the fractional parts separately.

$6\frac{3}{4} = 6 + \frac{3}{4}$

We need to calculate $\frac{3}{8} + 6\frac{3}{4} = \frac{3}{8} + (6 + \frac{3}{4})$

This can be written as $6 + (\frac{3}{8} + \frac{3}{4})$.

First, add the fractional parts: $\frac{3}{8} + \frac{3}{4}$.

The common denominator for 8 and 4 is 8.

Convert $\frac{3}{4}$ to an equivalent fraction with denominator 8:

$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$

Now add the fractions: $\frac{3}{8} + \frac{6}{8} = \frac{3+6}{8} = \frac{9}{8}$.

The sum of the fractional parts is $\frac{9}{8}$. This is an improper fraction, which can be written as a mixed fraction $1\frac{1}{8}$.

Now, add this sum to the whole number part (6).

$6 + 1\frac{1}{8} = 6 + 1 + \frac{1}{8} = 7 + \frac{1}{8} = 7\frac{1}{8}$.

Both methods yield the same result.

Final Answer:

The sum of $\frac{3}{8}$ and $6\frac{3}{4}$ is $\frac{57}{8}$ or $7\frac{1}{8}$.

Given:

The fractions are $\frac{1}{2}$ and $\frac{1}{6}$.

To Find:

The result of subtracting $\frac{1}{6}$ from $\frac{1}{2}$, which means calculating $\frac{1}{2} - \frac{1}{6}$.

Solution:

To subtract fractions with different denominators, we first find a common denominator, which is the least common multiple (LCM) of the denominators.

The denominators are 2 and 6.

The multiples of 2 are 2, 4, 6, 8, ...

The multiples of 6 are 6, 12, 18, ...

The least common multiple (LCM) of 2 and 6 is 6.

Now, we convert each fraction to an equivalent fraction with a denominator of 6.

For $\frac{1}{2}$: Multiply the numerator and denominator by $\frac{6}{2} = 3$.

$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

For $\frac{1}{6}$: The denominator is already 6, so it remains $\frac{1}{6}$.

Now that the fractions have the same denominator, we can subtract them by subtracting their numerators and keeping the common denominator.

$\frac{1}{2} - \frac{1}{6} = \frac{3}{6} - \frac{1}{6}$

Difference $= \frac{3 - 1}{6} = \frac{2}{6}$

The resulting fraction is $\frac{2}{6}$. We need to simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

The GCD of 2 and 6 is 2.

$\frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3}$

The simplified fraction is $\frac{1}{3}$.

Final Answer:

The result of subtracting $\frac{1}{6}$ from $\frac{1}{2}$ is $\frac{1}{3}$.

Given:

The numbers are $\frac{100}{9}$ and $8\frac{1}{3}$.

To Find:

The difference obtained by subtracting $8\frac{1}{3}$ from $\frac{100}{9}$, i.e., $\frac{100}{9} - 8\frac{1}{3}$.

Solution:

First, we convert the mixed fraction $8\frac{1}{3}$ into an improper fraction.

$8\frac{1}{3} = \frac{(8 \times 3) + 1}{3} = \frac{24 + 1}{3} = \frac{25}{3}$.

Now the problem is to calculate $\frac{100}{9} - \frac{25}{3}$.

To subtract fractions with different denominators, we find the least common multiple (LCM) of the denominators.

The denominators are 9 and 3.

The LCM of 9 and 3 is 9.

Next, we convert each fraction to an equivalent fraction with a denominator of 9.

The first fraction $\frac{100}{9}$ already has the denominator 9.

For the second fraction $\frac{25}{3}$, we multiply the numerator and the denominator by the factor needed to change 3 to 9, which is $9 \div 3 = 3$.

$\frac{25}{3} = \frac{25 \times 3}{3 \times 3} = \frac{75}{9}$.

Now we subtract the equivalent fractions:

$\frac{100}{9} - \frac{25}{3} = \frac{100}{9} - \frac{75}{9}$

Subtract the numerators and keep the common denominator:

Difference $= \frac{100 - 75}{9} = \frac{25}{9}$.

The result is an improper fraction $\frac{25}{9}$.

Final Answer:

The result of subtracting $8\frac{1}{3}$ from $\frac{100}{9}$ is $\frac{25}{9}$.

Given:

The numbers are $6\frac{1}{2}$ and $1\frac{1}{4}$. We need to calculate $6\frac{1}{2} - 1\frac{1}{4}$.

To Find:

The difference obtained by subtracting $1\frac{1}{4}$ from $6\frac{1}{2}$.

Solution:

Method 1: Convert mixed fractions to improper fractions.

Convert $6\frac{1}{2}$ to an improper fraction:

$6\frac{1}{2} = \frac{(6 \times 2) + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2}$

Convert $1\frac{1}{4}$ to an improper fraction:

$1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4}$

Now, subtract the improper fractions: $\frac{13}{2} - \frac{5}{4}$.

Find a common denominator for 2 and 4. The least common multiple (LCM) of 2 and 4 is 4.

Convert $\frac{13}{2}$ to an equivalent fraction with denominator 4:

$\frac{13}{2} = \frac{13 \times 2}{2 \times 2} = \frac{26}{4}$

Now subtract the fractions with the common denominator:

$\frac{26}{4} - \frac{5}{4} = \frac{26 - 5}{4} = \frac{21}{4}$

The result is the improper fraction $\frac{21}{4}$. To express this as a mixed fraction, divide the numerator (21) by the denominator (4).

$21 \div 4$. The quotient is 5 and the remainder is 1.

So, $\frac{21}{4} = 5\frac{1}{4}$.

Alternate Solution:

Method 2: Subtract whole numbers and fractions separately.

$6\frac{1}{2} - 1\frac{1}{4}$

Subtract the whole number parts: $6 - 1 = 5$.

Subtract the fractional parts: $\frac{1}{2} - \frac{1}{4}$.

Find a common denominator for $\frac{1}{2}$ and $\frac{1}{4}$. The LCM of 2 and 4 is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with denominator 4:

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

Now subtract the fractions: $\frac{2}{4} - \frac{1}{4} = \frac{2 - 1}{4} = \frac{1}{4}$.

Combine the difference of the whole numbers and the difference of the fractions:

$5 + \frac{1}{4} = 5\frac{1}{4}$.

Both methods give the same result.

Final Answer:

The result of subtracting $1\frac{1}{4}$ from $6\frac{1}{2}$ is $5\frac{1}{4}$.

Given:

The numbers to be added are the mixed fractions $1\frac{1}{4}$ and $6\frac{1}{2}$.

To Find:

The sum of $1\frac{1}{4}$ and $6\frac{1}{2}$.

Solution:

Method 1: Convert mixed fractions to improper fractions.

Convert $1\frac{1}{4}$ to an improper fraction:

$1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4}$

Convert $6\frac{1}{2}$ to an improper fraction:

$6\frac{1}{2} = \frac{(6 \times 2) + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2}$

Now, add the improper fractions: $\frac{5}{4} + \frac{13}{2}$.

To add fractions with different denominators, we find a common denominator. The denominators are 4 and 2.

The least common multiple (LCM) of 4 and 2 is 4.

Convert $\frac{13}{2}$ to an equivalent fraction with denominator 4:

$\frac{13}{2} = \frac{13 \times 2}{2 \times 2} = \frac{26}{4}$

Now add the fractions with the common denominator:

$\frac{5}{4} + \frac{26}{4} = \frac{5 + 26}{4} = \frac{31}{4}$

The result is the improper fraction $\frac{31}{4}$. To express this as a mixed fraction, divide the numerator (31) by the denominator (4).

$31 \div 4$. The quotient is 7 and the remainder is $31 - (4 \times 7) = 31 - 28 = 3$.

So, $\frac{31}{4} = 7\frac{3}{4}$.

Alternate Solution:

Method 2: Add whole numbers and fractions separately.

The sum is $(1 + \frac{1}{4}) + (6 + \frac{1}{2})$.

Group the whole numbers and the fractional parts:

$(1 + 6) + (\frac{1}{4} + \frac{1}{2})$

Add the whole numbers: $1 + 6 = 7$.

Add the fractional parts: $\frac{1}{4} + \frac{1}{2}$.

Find a common denominator for $\frac{1}{4}$ and $\frac{1}{2}$. The LCM of 4 and 2 is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with denominator 4:

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

Now add the fractions: $\frac{1}{4} + \frac{2}{4} = \frac{1 + 2}{4} = \frac{3}{4}$.

Combine the sum of the whole numbers and the sum of the fractions:

$7 + \frac{3}{4} = 7\frac{3}{4}$.

Both methods give the same result.

Final Answer:

The sum of $1\frac{1}{4}$ and $6\frac{1}{2}$ is $7\frac{3}{4}$.

Given:

Distance travelled in the morning = $6\frac{1}{2}$ km

Distance travelled in the evening = $8\frac{3}{4}$ km

To Find:

The total distance travelled by Katrina on that day.

Solution:

To find the total distance travelled, we need to add the distance travelled in the morning and the distance travelled in the evening.

Total distance = Distance in morning + Distance in evening

Total distance = $6\frac{1}{2} + 8\frac{3}{4}$ km

We can add these mixed fractions by converting them to improper fractions first.

Convert $6\frac{1}{2}$ to an improper fraction:

$6\frac{1}{2} = \frac{(6 \times 2) + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2}$

Convert $8\frac{3}{4}$ to an improper fraction:

$8\frac{3}{4} = \frac{(8 \times 4) + 3}{4} = \frac{32 + 3}{4} = \frac{35}{4}$

Now add the improper fractions:

Total distance = $\frac{13}{2} + \frac{35}{4}$

To add fractions with different denominators, find a common denominator. The denominators are 2 and 4. The least common multiple (LCM) of 2 and 4 is 4.

Convert $\frac{13}{2}$ to an equivalent fraction with denominator 4 by multiplying the numerator and denominator by 2:

$\frac{13}{2} = \frac{13 \times 2}{2 \times 2} = \frac{26}{4}$

Now add the fractions with the common denominator:

Total distance = $\frac{26}{4} + \frac{35}{4} = \frac{26 + 35}{4} = \frac{61}{4}$

The total distance is $\frac{61}{4}$ km. This is an improper fraction. We can convert it to a mixed fraction by dividing 61 by 4.

$61 \div 4$. The quotient is 15 and the remainder is 1 ($61 = 4 \times 15 + 1$).

So, $\frac{61}{4} = 15\frac{1}{4}$.

Alternate Solution:

Add the whole number parts and the fractional parts separately.

$6\frac{1}{2} + 8\frac{3}{4} = (6 + \frac{1}{2}) + (8 + \frac{3}{4})$

Group the whole numbers and the fractional parts:

$= (6 + 8) + (\frac{1}{2} + \frac{3}{4})$

Add the whole numbers: $6 + 8 = 14$.

Add the fractional parts: $\frac{1}{2} + \frac{3}{4}$. The common denominator for 2 and 4 is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with denominator 4: $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

Add the fractions: $\frac{2}{4} + \frac{3}{4} = \frac{2 + 3}{4} = \frac{5}{4}$.

The sum of the fractional parts is $\frac{5}{4}$. This is an improper fraction, which is equal to $1\frac{1}{4}$.

Combine the sum of the whole numbers and the sum of the fractional parts:

$14 + 1\frac{1}{4} = 14 + 1 + \frac{1}{4} = 15 + \frac{1}{4} = 15\frac{1}{4}$.

Both methods give the same result.

Final Answer:

The total distance travelled by Katrina altogether on that day is $15\frac{1}{4}$ km.

Given:

A rectangle is divided into an unknown number of equal parts.

16 of these parts represent the fraction $\frac{1}{4}$ of the whole rectangle.

To Find:

The total number of equal parts into which the rectangle has been divided.

Solution:

Let the total number of equal parts into which the rectangle is divided be $N$.

Each part represents the fraction $\frac{1}{N}$ of the whole rectangle.

We are given that 16 of these parts represent the fraction $\frac{1}{4}$.

The fraction represented by 16 parts is $16 \times (\text{fraction represented by one part})$.

Fraction represented by 16 parts = $16 \times \frac{1}{N} = \frac{16}{N}$.

According to the problem statement, this fraction is equal to $\frac{1}{4}$.

So, we can set up the equation:

To solve for $N$, we can cross-multiply the equation (i).

$16 \times 4 = N \times 1$

$64 = N$

$N = 64$

Thus, the total number of parts in which the rectangle has been divided is 64.

We can verify this: If there are 64 parts, then $\frac{1}{4}$ of the parts is $\frac{1}{4} \times 64 = 16$ parts. This matches the given information.

Final Answer:

The number of parts in which the rectangle has been divided is 64.

Given:

The grip size of a tennis racquet is $11\frac{9}{80}$ cm.

To Express:

Express the given mixed fraction as an improper fraction.

Solution:

The given mixed fraction is $11\frac{9}{80}$.

A mixed fraction in the form $a\frac{b}{c}$ can be converted to an improper fraction using the formula:

Improper fraction = $\frac{(\text{Whole number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}$

In the mixed fraction $11\frac{9}{80}$, the whole number part is 11, the numerator is 9, and the denominator is 80.

Applying the formula:

$11\frac{9}{80} = \frac{(11 \times 80) + 9}{80}$

First, calculate the product of the whole number and the denominator:

$11 \times 80 = 880$

Now, add the numerator to this product:

$880 + 9 = 889$

The improper fraction is $\frac{\text{New Numerator}}{\text{Original Denominator}}$.

Improper fraction = $\frac{889}{80}$

We should check if the fraction $\frac{889}{80}$ can be simplified. The prime factors of 80 are $2^4 \times 5$. We need to check if 889 is divisible by 2 or 5. 889 is not divisible by 2 (it's odd) and does not end in 0 or 5 (not divisible by 5). We could check for other prime factors, but for this level, it's usually sufficient to check the prime factors of the denominator. 889 is not easily divisible by small primes, and it turns out to be prime itself. So, the fraction is in its simplest form.

Final Answer:

The grip size $11\frac{9}{80}$ cm expressed as an improper fraction is $\frac{889}{80}$ cm.

Given:

Fraction of food eaten that is available for the next level = $\frac{1}{10}$

To Find:

The fraction of the food eaten that is not available for the next level.

Solution:

Let the total fraction of food eaten be represented by 1 (or $\frac{10}{10}$).

The food eaten is either turned into the organism's own body (available for the next level) or it is used for energy, lost as waste, etc. (not available for the next level).

Fraction of food eaten = Fraction available for next level + Fraction not available for next level

Let $F_{available}$ be the fraction of food available for the next level, and $F_{not\ available}$ be the fraction of food not available for the next level.

Total food eaten (as a fraction) = 1

We are given $F_{available} = \frac{1}{10}$.

So, $1 = F_{available} + F_{not\ available}$

Substituting the given value:

$1 = \frac{1}{10} + F_{not\ available}$

To find the fraction not available, we subtract the fraction available from the total food eaten (1).

$F_{not\ available} = 1 - \frac{1}{10}$

To subtract the fraction from 1, we express 1 as a fraction with the same denominator as the given fraction.

$1 = \frac{10}{10}$

So, $F_{not\ available} = \frac{10}{10} - \frac{1}{10}$

Subtract the numerators and keep the common denominator:

$F_{not\ available} = \frac{10 - 1}{10} = \frac{9}{10}$

Thus, $\frac{9}{10}$ of the food eaten is not available for the next level of consumer.

Final Answer:

The fraction of the food eaten that is not available for the next level is $\frac{9}{10}$.

Given:

Age when Mr. Rajan got the job = 24 years

Age when Mr. Rajan retired = 60 years

To Find:

The fraction of his age till retirement that he was in the job.

Solution:

First, we need to find the duration Mr. Rajan was in the job.

Duration in job = Age at retirement - Age when job started

Duration in job = 60 years - 24 years = 36 years

The total age considered for the fraction is his age till retirement, which is 60 years.

The fraction of his age till retirement that he was in the job is given by:

Fraction = $\frac{\text{Duration in job}}{\text{Age till retirement}}$

Fraction = $\frac{36 \text{ years}}{60 \text{ years}} = \frac{36}{60}$

Now, we need to simplify the fraction $\frac{36}{60}$. We find the greatest common divisor (GCD) of 36 and 60.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

The greatest common divisor of 36 and 60 is 12.

Divide both the numerator and the denominator by 12:

$\frac{36}{60} = \frac{36 \div 12}{60 \div 12} = \frac{3}{5}$

The simplified fraction is $\frac{3}{5}$.

Final Answer:

Mr. Rajan was in the job for $\frac{3}{5}$ fraction of his age till retirement.

Given:

Maximum time food remains in the stomach = 4 hours

To Find:

The fraction of a day for which the food remains in the stomach.

Solution:

To find the fraction of a day, we need to compare the time the food remains in the stomach with the total number of hours in a day.

We know that the number of hours in a day is 24.

Number of hours food remains in stomach = 4 hours

Total hours in a day = 24 hours

The fraction of a day is given by:

Fraction = $\frac{\text{Time in stomach}}{\text{Total time in a day}}$

Fraction = $\frac{4 \text{ hours}}{24 \text{ hours}} = \frac{4}{24}$

Now, we need to simplify the fraction $\frac{4}{24}$. We find the greatest common divisor (GCD) of 4 and 24.

The factors of 4 are 1, 2, 4.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

The greatest common divisor of 4 and 24 is 4.

Divide both the numerator and the denominator by 4:

$\frac{4}{24} = \frac{4 \div 4}{24 \div 4} = \frac{1}{6}$

The simplified fraction is $\frac{1}{6}$.

Final Answer:

The food remains in the stomach for $\frac{1}{6}$ fraction of a day.

Given:

We are given a number 25.5.

We want to find a number that, when added to 25.5, results in 50.

To Find:

The number that should be added to 25.5 to get 50.

Solution:

Let the unknown number be $x$.

According to the problem statement, when $x$ is added to 25.5, the result is 50.

We can write this as an equation:

To find the value of $x$, we need to isolate $x$ on one side of the equation. We can do this by subtracting 25.5 from both sides of the equation (i).

$x = 50 - 25.5$

Now, we perform the subtraction of the decimal numbers.

We can write 50 as 50.0 to align the decimal points for subtraction.

$50.0 - 25.5$

Starting from the rightmost digit (tenths place): $0 - 5$. We need to borrow from the units place. The 0 in the units place becomes 10 (after borrowing 1 from the tens place), and the 5 in the tens place becomes 4.

$10 - 5 = 5$. Write 5 in the tenths place.

In the units place: $9 - 5 = 4$ (since we borrowed 1 from the tens place, the original 0 in units became 10, and after borrowing to the tenths place it became 9). Write 4 in the units place.

In the tens place: $4 - 2 = 2$ (since we borrowed 1 from the tens place). Write 2 in the tens place.

Place the decimal point in the result in line with the decimal points in the numbers being subtracted.

So, $50 - 25.5 = 24.5$.

Therefore, $x = 24.5$.

We can verify the answer by adding 25.5 and 24.5:

$25.5 + 24.5 = 50.0$

Final Answer:

24.5 should be added to 25.5 to get 50.

Given:

Weight of potatoes = 1 kg 200 g

Weight of dhania = 250 g

Weight of onion = 5 kg 300 g

Weight of palak = 500 g

Weight of tomatoes = 2 kg 600 g

To Find:

The total weight of Alok's purchases in kilograms.

Solution:

To find the total weight in kilograms, we need to convert each weight to kilograms and then add them together.

We know that 1 kilogram (kg) = 1000 grams (g).

Therefore, 1 gram (g) = $\frac{1}{1000}$ kilograms (kg).

Converting each item's weight to kilograms:

Weight of potatoes = $1 \text{ kg } 200 \text{ g} = 1 \text{ kg} + 200 \times \frac{1}{1000} \text{ kg} = 1 \text{ kg} + \frac{200}{1000} \text{ kg} = 1 \text{ kg} + 0.200 \text{ kg} = 1.200 \text{ kg}$.

Weight of dhania = $250 \text{ g} = 250 \times \frac{1}{1000} \text{ kg} = \frac{250}{1000} \text{ kg} = 0.250 \text{ kg}$.

Weight of onion = $5 \text{ kg } 300 \text{ g} = 5 \text{ kg} + 300 \times \frac{1}{1000} \text{ kg} = 5 \text{ kg} + \frac{300}{1000} \text{ kg} = 5 \text{ kg} + 0.300 \text{ kg} = 5.300 \text{ kg}$.

Weight of palak = $500 \text{ g} = 500 \times \frac{1}{1000} \text{ kg} = \frac{500}{1000} \text{ kg} = 0.500 \text{ kg}$.

Weight of tomatoes = $2 \text{ kg } 600 \text{ g} = 2 \text{ kg} + 600 \times \frac{1}{1000} \text{ kg} = 2 \text{ kg} + \frac{600}{1000} \text{ kg} = 2 \text{ kg} + 0.600 \text{ kg} = 2.600 \text{ kg}$.

Now, add all these weights to find the total weight.

Total weight = $1.200 \text{ kg} + 0.250 \text{ kg} + 5.300 \text{ kg} + 0.500 \text{ kg} + 2.600 \text{ kg}$.

Adding the decimal numbers:

The sum is 9.850 kg.

Therefore, the total weight of Alok's purchases is 9.850 kg.

Final Answer:

The total weight of Alok’s purchases in kilograms is 9.850 kg.

Given:

The decimal numbers are: 0.011, 1.001, 0.101, 0.110.

To Arrange:

Arrange the given decimal numbers in ascending order (from smallest to largest).

Solution:

To arrange decimal numbers in ascending order, we compare them digit by digit from left to right, starting with the whole number part.

The whole number parts of the given numbers are:

0.011 -> 0

1.001 -> 1

0.101 -> 0

0.110 -> 0

The smallest whole number part is 0. The numbers with the whole number part 0 are 0.011, 0.101, and 0.110. The number with the whole number part 1 (1.001) is the largest.

Now, let's compare the numbers with the whole number part 0 (0.011, 0.101, 0.110) by looking at the digits after the decimal point, starting from the tenths place.

Comparing the tenths digits:

0.011 (tenths digit is 0)

0.101 (tenths digit is 1)

0.110 (tenths digit is 1)

The smallest tenths digit is 0 (in 0.011). So, 0.011 is the smallest number among these three.

Next, compare the numbers with the tenths digit 1: 0.101 and 0.110. Look at the hundredths digit.

Comparing the hundredths digits:

0.101 (hundredths digit is 0)

0.110 (hundredths digit is 1)

The smallest hundredths digit is 0 (in 0.101). So, 0.101 is smaller than 0.110.

The hundredths digit in 0.110 is 1, which is larger than 0.

So, the order of the numbers starting with 0, from smallest to largest, is 0.011, 0.101, 0.110.

The largest number overall is 1.001.

Combining all numbers in ascending order:

0.011, 0.101, 0.110, 1.001.

Final Answer:

The numbers arranged in ascending order are:

0.011, 0.101, 0.110, 1.001.

Given:

The decimal numbers to be added are 20.02 and 2.002.

To Find:

The sum of 20.02 and 2.002.

Solution:

To add decimal numbers, we align the decimal points and add the numbers as we would add whole numbers, starting from the rightmost digit.

We can write the numbers one below the other, ensuring the decimal points are vertically aligned. We can add trailing zeros to the number with fewer decimal places so that both numbers have the same number of decimal places.

20.02 has two decimal places.

2.002 has three decimal places.

Write 20.02 as 20.020 to have three decimal places.

Now, perform the addition:

20.020 + 2.002

Adding the numbers:

Starting from the rightmost column (thousandths place): $0 + 2 = 2$. Write 2.

Next column (hundredths place): $2 + 0 = 2$. Write 2.

Next column (tenths place): $0 + 0 = 0$. Write 0.

Place the decimal point.

Next column (units place): $0 + 2 = 2$. Write 2.

Next column (tens place): $2 + 0 = 2$. Write 2.

The sum is 22.022.

Final Answer:

The sum of 20.02 and 2.002 is 22.022.

Given:

Quantity of CNG saved = 33000 tonnes

Quantity of diesel saved = 3300 tonnes

Quantity of petrol saved = 21000 tonnes

To Find:

(i) The fraction of the quantity of diesel saved to the quantity of petrol saved.

(ii) The fraction of the quantity of diesel saved to the quantity of CNG saved.

Solution:

(i) Fraction of the quantity of diesel saved to the quantity of petrol saved.

This fraction is given by:

Fraction = $\frac{\text{Quantity of diesel saved}}{\text{Quantity of petrol saved}}$

Substitute the given values:

Fraction = $\frac{3300 \text{ tonnes}}{21000 \text{ tonnes}} = \frac{3300}{21000}$

Now, simplify the fraction. We can cancel out the common factors from the numerator and the denominator.

First, cancel out the trailing zeros (which is equivalent to dividing both by 100):

$\frac{3300}{21000} = \frac{33 \times \cancel{100}}{210 \times \cancel{100}} = \frac{33}{210}$

Now, find the greatest common divisor (GCD) of 33 and 210. Both are divisible by 3.

$\frac{33}{210} = \frac{33 \div 3}{210 \div 3} = \frac{11}{70}$

The simplified fraction is $\frac{11}{70}$.

(ii) Fraction of the quantity of diesel saved to the quantity of CNG saved.

This fraction is given by:

Fraction = $\frac{\text{Quantity of diesel saved}}{\text{Quantity of CNG saved}}$

Substitute the given values:

Fraction = $\frac{3300 \text{ tonnes}}{33000 \text{ tonnes}} = \frac{3300}{33000}$

Now, simplify the fraction. Cancel out the common factors.

First, cancel out the trailing zeros (dividing by 100):

$\frac{3300}{33000} = \frac{33 \times \cancel{100}}{330 \times \cancel{100}} = \frac{33}{330}$

Now, divide both numerator and denominator by 33:

$\frac{33}{330} = \frac{33 \div 33}{330 \div 33} = \frac{1}{10}$

The simplified fraction is $\frac{1}{10}$.

Final Answer:

(i) The fraction of the quantity of diesel saved to the quantity of petrol saved is $\frac{11}{70}$.

(ii) The fraction of the quantity of diesel saved to the quantity of CNG saved is $\frac{1}{10}$.

Given:

Energy content per kg for different foods as provided in the table:

Wheat: 3.2 Joules/kg

Rice: 5.3 Joules/kg

Potatoes (Cooked): 3.7 Joules/kg

Milk: 3.0 Joules/kg

To Find:

1. The food that provides the least energy.

2. The food that provides the maximum energy.

3. The fraction of the least energy to the maximum energy.

Solution:

We compare the energy content values for each food:

3.2, 5.3, 3.7, 3.0

To find the least energy, we look for the smallest value among these numbers. The smallest value is 3.0, which corresponds to Milk.

The food that provides the least energy is Milk (3.0 Joules/kg).

To find the maximum energy, we look for the largest value among these numbers. The largest value is 5.3, which corresponds to Rice.

The food that provides the maximum energy is Rice (5.3 Joules/kg).

Now, we need to express the least energy as a fraction of the maximum energy.

Least energy = 3.0 Joules/kg

Maximum energy = 5.3 Joules/kg

Fraction = $\frac{\text{Least Energy}}{\text{Maximum Energy}}$

Fraction = $\frac{3.0}{5.3}$

To express this as a fraction without decimals, we can multiply both the numerator and the denominator by 10.

Fraction = $\frac{3.0 \times 10}{5.3 \times 10} = \frac{30}{53}$

We check if the fraction $\frac{30}{53}$ can be simplified. The number 53 is a prime number. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. None of these factors (except 1) are 53. Therefore, the fraction is in its simplest form.

Final Answer:

The food that provides the least energy is Milk.

The food that provides the maximum energy is Rice.

The fraction of the least energy to the maximum energy is $\frac{30}{53}$.

Given:

The cup is $\frac{1}{3}$ full of milk.

To Find:

The part of the cup that needs to be filled to make it full.

Solution:

A full cup can be represented by the fraction 1 (or $\frac{3}{3}$ in this case, since the denominator of the given fraction is 3).

The part of the cup that is already filled is $\frac{1}{3}$.

The part of the cup that is still to be filled is the difference between a full cup (1) and the part that is already filled ($\frac{1}{3}$).

Part to be filled = Full cup - Part filled

Part to be filled = $1 - \frac{1}{3}$

To subtract the fraction from 1, we express 1 as a fraction with the same denominator as the other fraction, which is 3.

$1 = \frac{3}{3}$

So, Part to be filled = $\frac{3}{3} - \frac{1}{3}$

Now, subtract the numerators and keep the common denominator:

Part to be filled = $\frac{3 - 1}{3} = \frac{2}{3}$

Thus, $\frac{2}{3}$ part of the cup is still to be filled with milk to make it full.

Final Answer:

The part of the cup that is still to be filled by milk is $\frac{2}{3}$.

Given:

Total length of lace bought by Mary = $3\frac{1}{2}$ m

Length of lace used for the dress = $1\frac{3}{4}$ m

To Find:

The length of lace left with Mary.

Solution:

To find the length of lace left, we need to subtract the length of lace used from the total length of lace bought.

Lace left = Total lace bought - Lace used

Lace left = $3\frac{1}{2} - 1\frac{3}{4}$ m

We can solve this by converting the mixed fractions to improper fractions.

Convert $3\frac{1}{2}$ to an improper fraction:

$3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}$

Convert $1\frac{3}{4}$ to an improper fraction:

$1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}$

Now subtract the improper fractions: $\frac{7}{2} - \frac{7}{4}$.

To subtract fractions with different denominators, find a common denominator. The denominators are 2 and 4. The least common multiple (LCM) of 2 and 4 is 4.

Convert $\frac{7}{2}$ to an equivalent fraction with denominator 4 by multiplying the numerator and denominator by 2:

$\frac{7}{2} = \frac{7 \times 2}{2 \times 2} = \frac{14}{4}$

Now subtract the fractions with the common denominator:

Lace left = $\frac{14}{4} - \frac{7}{4} = \frac{14 - 7}{4} = \frac{7}{4}$

The result is the improper fraction $\frac{7}{4}$ m. We can convert this to a mixed fraction by dividing 7 by 4.

$7 \div 4$. The quotient is 1 and the remainder is $7 - (4 \times 1) = 7 - 4 = 3$.

So, $\frac{7}{4} = 1\frac{3}{4}$.

Alternate Solution:

Subtract the whole number parts and the fractional parts separately.

$6\frac{1}{2} - 1\frac{3}{4}$

We can write $3\frac{1}{2}$ as $3 + \frac{1}{2}$ and $1\frac{3}{4}$ as $1 + \frac{3}{4}$.

The subtraction is $(3 + \frac{1}{2}) - (1 + \frac{3}{4})$.

This is equal to $(3 - 1) + (\frac{1}{2} - \frac{3}{4})$.

Subtract the whole numbers: $3 - 1 = 2$.

Subtract the fractional parts: $\frac{1}{2} - \frac{3}{4}$. Find a common denominator (LCM of 2 and 4 is 4).

Convert $\frac{1}{2}$ to $\frac{2}{4}$.

Now subtract: $\frac{2}{4} - \frac{3}{4}$.

Since $\frac{2}{4} < \frac{3}{4}$, we need to borrow from the whole number part. We borrow 1 from the whole number 2, which is equal to $\frac{4}{4}$. Add this to the fractional part $\frac{2}{4}$.

The expression becomes $1 + (\frac{4}{4} + \frac{2}{4}) - \frac{3}{4} = 1 + \frac{6}{4} - \frac{3}{4}$.

Subtract the fractional parts: $\frac{6}{4} - \frac{3}{4} = \frac{6 - 3}{4} = \frac{3}{4}$.

Combine the whole number part (1) and the fractional part ($\frac{3}{4}$).

Result $= 1\frac{3}{4}$.

Both methods yield the same result.

Final Answer:

The length of lace left with Mary is $\frac{7}{4}$ m or $1\frac{3}{4}$ m.

Given:

Earlier weight of Sunita = $46\frac{3}{8}$ kg

Weight gained on Monday = $1\frac{1}{4}$ kg

To Find:

Sunita's weight on Monday.

Solution:

To find Sunita's weight on Monday, we need to add the weight she gained to her earlier weight.

Weight on Monday = Earlier weight + Weight gained

Weight on Monday = $46\frac{3}{8} \text{ kg } + 1\frac{1}{4} \text{ kg}$

We can add these mixed fractions by first converting them to improper fractions.

Convert $46\frac{3}{8}$ to an improper fraction:

$46\frac{3}{8} = \frac{(46 \times 8) + 3}{8} = \frac{368 + 3}{8} = \frac{371}{8}$

Convert $1\frac{1}{4}$ to an improper fraction:

$1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4}$

Now add the improper fractions: $\frac{371}{8} + \frac{5}{4}$.

To add fractions with different denominators, find a common denominator. The denominators are 8 and 4. The least common multiple (LCM) of 8 and 4 is 8.

Convert $\frac{5}{4}$ to an equivalent fraction with denominator 8 by multiplying the numerator and denominator by $\frac{8}{4} = 2$:

$\frac{5}{4} = \frac{5 \times 2}{4 \times 2} = \frac{10}{8}$

Now add the fractions with the common denominator:

Weight on Monday = $\frac{371}{8} + \frac{10}{8} = \frac{371 + 10}{8} = \frac{381}{8}$

The weight on Monday is $\frac{381}{8}$ kg. This is an improper fraction. We can convert it to a mixed fraction by dividing 381 by 8.

$381 \div 8$. $381 = (47 \times 8) + 5$.

The quotient is 47 and the remainder is 5.

So, $\frac{381}{8} = 47\frac{5}{8}$.

Alternate Solution:

Add the whole number parts and the fractional parts separately.

Sum = $46\frac{3}{8} + 1\frac{1}{4} = (46 + \frac{3}{8}) + (1 + \frac{1}{4})$

Group the whole numbers and the fractional parts:

$= (46 + 1) + (\frac{3}{8} + \frac{1}{4})$

Add the whole numbers: $46 + 1 = 47$.

Add the fractional parts: $\frac{3}{8} + \frac{1}{4}$. The common denominator for 8 and 4 is 8.

Convert $\frac{1}{4}$ to an equivalent fraction with denominator 8: $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$.

Add the fractions: $\frac{3}{8} + \frac{2}{8} = \frac{3 + 2}{8} = \frac{5}{8}$.

Combine the sum of the whole numbers and the sum of the fractional parts:

Result $= 47 + \frac{5}{8} = 47\frac{5}{8}$.

Both methods give the same result.

Final Answer:

Sunita's weight on Monday was $47\frac{5}{8}$ kg.

Given:

Quantity of juice purchased on Monday = $12\frac{1}{2}$ litres

Quantity of juice purchased on Tuesday = $14\frac{3}{4}$ litres

To Find:

The total quantity of juice purchased in two days.

Solution:

To find the total quantity of juice purchased, we need to add the quantities purchased on Monday and Tuesday.

Total quantity = Quantity on Monday + Quantity on Tuesday

Total quantity = $12\frac{1}{2} + 14\frac{3}{4}$ litres

Method 1: Convert mixed fractions to improper fractions.

Convert $12\frac{1}{2}$ to an improper fraction:

$12\frac{1}{2} = \frac{(12 \times 2) + 1}{2} = \frac{24 + 1}{2} = \frac{25}{2}$

Convert $14\frac{3}{4}$ to an improper fraction:

$14\frac{3}{4} = \frac{(14 \times 4) + 3}{4} = \frac{56 + 3}{4} = \frac{59}{4}$

Now, add the improper fractions: $\frac{25}{2} + \frac{59}{4}$.

To add fractions with different denominators, find a common denominator. The denominators are 2 and 4. The least common multiple (LCM) of 2 and 4 is 4.

Convert $\frac{25}{2}$ to an equivalent fraction with denominator 4 by multiplying the numerator and denominator by $\frac{4}{2} = 2$:

$\frac{25}{2} = \frac{25 \times 2}{2 \times 2} = \frac{50}{4}$

Now add the fractions with the common denominator:

Total quantity = $\frac{50}{4} + \frac{59}{4} = \frac{50 + 59}{4} = \frac{109}{4}$

The total quantity is $\frac{109}{4}$ litres. This is an improper fraction. We can convert it to a mixed fraction by dividing 109 by 4.

$109 \div 4$. $109 = (4 \times 27) + 1$.

The quotient is 27 and the remainder is 1.

So, $\frac{109}{4} = 27\frac{1}{4}$.

Alternate Solution:

Method 2: Add whole numbers and fractions separately.

The sum is $12\frac{1}{2} + 14\frac{3}{4} = (12 + \frac{1}{2}) + (14 + \frac{3}{4})$

Group the whole numbers and the fractional parts:

$= (12 + 14) + (\frac{1}{2} + \frac{3}{4})$

Add the whole numbers: $12 + 14 = 26$.

Add the fractional parts: $\frac{1}{2} + \frac{3}{4}$. The common denominator for 2 and 4 is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with denominator 4: $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

Add the fractions: $\frac{2}{4} + \frac{3}{4} = \frac{2 + 3}{4} = \frac{5}{4}$.

The sum of the fractional parts is $\frac{5}{4}$. This is an improper fraction, which is equal to $1\frac{1}{4}$.

Combine the sum of the whole numbers and the sum of the fractional parts:

Result $= 26 + 1\frac{1}{4} = 26 + 1 + \frac{1}{4} = 27 + \frac{1}{4} = 27\frac{1}{4}$.

Both methods yield the same result.

Final Answer:

Sunil purchased a total of $27\frac{1}{4}$ litres of juice in two days.

Given:

Total quantity of juice Nazima purchased = $5\frac{1}{2}$ litres

Quantity of juice given to friends = $2\frac{3}{4}$ litres

To Find:

The quantity of juice left with Nazima.

Solution:

To find the quantity of juice left, we need to subtract the quantity given to friends from the total quantity purchased.

Juice left = Total juice purchased - Juice given away

Juice left = $5\frac{1}{2} - 2\frac{3}{4}$ litres

Method 1: Convert mixed fractions to improper fractions.

Convert $5\frac{1}{2}$ to an improper fraction:

$5\frac{1}{2} = \frac{(5 \times 2) + 1}{2} = \frac{10 + 1}{2} = \frac{11}{2}$

Convert $2\frac{3}{4}$ to an improper fraction:

$2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$

Now subtract the improper fractions: $\frac{11}{2} - \frac{11}{4}$.

To subtract fractions with different denominators, find a common denominator. The denominators are 2 and 4. The least common multiple (LCM) of 2 and 4 is 4.

Convert $\frac{11}{2}$ to an equivalent fraction with denominator 4 by multiplying the numerator and denominator by $\frac{4}{2} = 2$:

$\frac{11}{2} = \frac{11 \times 2}{2 \times 2} = \frac{22}{4}$

Now subtract the fractions with the common denominator:

Juice left = $\frac{22}{4} - \frac{11}{4} = \frac{22 - 11}{4} = \frac{11}{4}$

The result is the improper fraction $\frac{11}{4}$ litres. We can convert this to a mixed fraction by dividing 11 by 4.

$11 \div 4$. The quotient is 2 and the remainder is $11 - (4 \times 2) = 11 - 8 = 3$.

So, $\frac{11}{4} = 2\frac{3}{4}$.

Alternate Solution:

Method 2: Subtract whole numbers and fractions separately.

The subtraction is $5\frac{1}{2} - 2\frac{3}{4}$.

We can write $5\frac{1}{2}$ as $5 + \frac{1}{2}$ and $2\frac{3}{4}$ as $2 + \frac{3}{4}$.

The expression is $(5 + \frac{1}{2}) - (2 + \frac{3}{4})$.

This is equal to $(5 - 2) + (\frac{1}{2} - \frac{3}{4})$.

Subtract the whole numbers: $5 - 2 = 3$.

Subtract the fractional parts: $\frac{1}{2} - \frac{3}{4}$. Find a common denominator (LCM of 2 and 4 is 4).

Convert $\frac{1}{2}$ to $\frac{2}{4}$.

Now subtract: $\frac{2}{4} - \frac{3}{4}$.

Since $\frac{2}{4} < \frac{3}{4}$, we need to borrow from the whole number part (3). We borrow 1 from 3, which is equal to $\frac{4}{4}$. The whole number part becomes 2. Add the borrowed $\frac{4}{4}$ to the fractional part $\frac{2}{4}$.

The expression becomes $2 + (\frac{4}{4} + \frac{2}{4}) - \frac{3}{4} = 2 + \frac{6}{4} - \frac{3}{4}$.

Subtract the fractional parts: $\frac{6}{4} - \frac{3}{4} = \frac{6 - 3}{4} = \frac{3}{4}$.

Combine the remaining whole number part (2) and the difference of the fractional parts ($\frac{3}{4}$).

Result $= 2\frac{3}{4}$.

Both methods yield the same result.

Final Answer:

Nazima is left with $\frac{11}{4}$ litres or $2\frac{3}{4}$ litres of juice.

Given:

Original length of the wooden board = $150\frac{1}{4}$ cm

Length of the piece sawed off = $40\frac{1}{5}$ cm

To Find:

The length of the remaining piece of the wooden board.

Solution:

To find the length of the remaining piece, we need to subtract the length of the piece sawed off from the original length of the board.

Length of remaining piece = Original length - Length sawed off

Length of remaining piece = $150\frac{1}{4} - 40\frac{1}{5}$ cm

Method 1: Convert mixed fractions to improper fractions.

Convert $150\frac{1}{4}$ to an improper fraction:

$150\frac{1}{4} = \frac{(150 \times 4) + 1}{4} = \frac{600 + 1}{4} = \frac{601}{4}$

Convert $40\frac{1}{5}$ to an improper fraction:

$40\frac{1}{5} = \frac{(40 \times 5) + 1}{5} = \frac{200 + 1}{5} = \frac{201}{5}$

Now subtract the improper fractions: $\frac{601}{4} - \frac{201}{5}$.

To subtract fractions with different denominators, find a common denominator. The denominators are 4 and 5. The least common multiple (LCM) of 4 and 5 is $4 \times 5 = 20$ (since they are relatively prime).

Convert $\frac{601}{4}$ to an equivalent fraction with denominator 20 by multiplying the numerator and denominator by $\frac{20}{4} = 5$:

$\frac{601}{4} = \frac{601 \times 5}{4 \times 5} = \frac{3005}{20}$

Convert $\frac{201}{5}$ to an equivalent fraction with denominator 20 by multiplying the numerator and denominator by $\frac{20}{5} = 4$:

$\frac{201}{5} = \frac{201 \times 4}{5 \times 4} = \frac{804}{20}$

Now subtract the fractions with the common denominator:

Length of remaining piece = $\frac{3005}{20} - \frac{804}{20} = \frac{3005 - 804}{20} = \frac{2201}{20}$

The result is the improper fraction $\frac{2201}{20}$ cm. We can convert this to a mixed fraction by dividing 2201 by 20.

$2201 \div 20$. $2201 = (110 \times 20) + 1$.

The quotient is 110 and the remainder is 1.

So, $\frac{2201}{20} = 110\frac{1}{20}$.

Alternate Solution:

Method 2: Subtract whole numbers and fractions separately.

The subtraction is $150\frac{1}{4} - 40\frac{1}{5} = (150 + \frac{1}{4}) - (40 + \frac{1}{5})$.

This is equal to $(150 - 40) + (\frac{1}{4} - \frac{1}{5})$.

Subtract the whole numbers: $150 - 40 = 110$.

Subtract the fractional parts: $\frac{1}{4} - \frac{1}{5}$. Find a common denominator (LCM of 4 and 5 is 20).

Convert $\frac{1}{4}$ to an equivalent fraction with denominator 20: $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$.

Convert $\frac{1}{5}$ to an equivalent fraction with denominator 20: $\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}$.

Now subtract the fractions: $\frac{5}{20} - \frac{4}{20} = \frac{5 - 4}{20} = \frac{1}{20}$.

Combine the difference of the whole numbers and the difference of the fractional parts:

Result $= 110 + \frac{1}{20} = 110\frac{1}{20}$.

Both methods yield the same result.

Final Answer:

The length of the remaining piece is $\frac{2201}{20}$ cm or $110\frac{1}{20}$ cm.

Given:

Distance travelled by bus = $3\frac{1}{2}$ km

Distance walked = $1\frac{1}{8}$ km

To Find:

The total distance travelled by Nasir to reach the town.

Solution:

To find the total distance travelled, we need to add the distance travelled by bus and the distance walked.

Total distance = Distance by bus + Distance walked

Total distance = $3\frac{1}{2} + 1\frac{1}{8}$ km

Method 1: Convert mixed fractions to improper fractions.

Convert $3\frac{1}{2}$ to an improper fraction:

$3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}$

Convert $1\frac{1}{8}$ to an improper fraction:

$1\frac{1}{8} = \frac{(1 \times 8) + 1}{8} = \frac{8 + 1}{8} = \frac{9}{8}$

Now, add the improper fractions: $\frac{7}{2} + \frac{9}{8}$.

To add fractions with different denominators, find a common denominator. The denominators are 2 and 8. The least common multiple (LCM) of 2 and 8 is 8.

Convert $\frac{7}{2}$ to an equivalent fraction with denominator 8 by multiplying the numerator and denominator by $\frac{8}{2} = 4$:

$\frac{7}{2} = \frac{7 \times 4}{2 \times 4} = \frac{28}{8}$

Now add the fractions with the common denominator:

Total distance = $\frac{28}{8} + \frac{9}{8} = \frac{28 + 9}{8} = \frac{37}{8}$

The total distance is $\frac{37}{8}$ km. This is an improper fraction. We can convert it to a mixed fraction by dividing 37 by 8.

$37 \div 8$. The quotient is 4 and the remainder is $37 - (8 \times 4) = 37 - 32 = 5$.

So, $\frac{37}{8} = 4\frac{5}{8}$.

Alternate Solution:

Method 2: Add whole numbers and fractions separately.

The sum is $3\frac{1}{2} + 1\frac{1}{8} = (3 + \frac{1}{2}) + (1 + \frac{1}{8})$

Group the whole numbers and the fractional parts:

$= (3 + 1) + (\frac{1}{2} + \frac{1}{8})$

Add the whole numbers: $3 + 1 = 4$.

Add the fractional parts: $\frac{1}{2} + \frac{1}{8}$. The common denominator for 2 and 8 is 8.

Convert $\frac{1}{2}$ to an equivalent fraction with denominator 8: $\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$.

Add the fractions: $\frac{4}{8} + \frac{1}{8} = \frac{4 + 1}{8} = \frac{5}{8}$.

Combine the sum of the whole numbers and the sum of the fractional parts:

Result $= 4 + \frac{5}{8} = 4\frac{5}{8}$.

Both methods yield the same result.

Final Answer:

Nasir travelled a total of $\frac{37}{8}$ km or $4\frac{5}{8}$ km to reach the town.

Given:

Weight of fish caught by Neetu = $3\frac{3}{4}$ kg

Weight of fish caught by Narendra = $2\frac{1}{2}$ kg

To Find:

How much more Neetu's fish weighed than Narendra's fish.

Solution:

To find out how much more Neetu's fish weighed, we need to subtract the weight of Narendra's fish from the weight of Neetu's fish.

Difference in weight = Weight of Neetu's fish - Weight of Narendra's fish

Difference in weight = $3\frac{3}{4} - 2\frac{1}{2}$ kg

Method 1: Convert mixed fractions to improper fractions.

Convert $3\frac{3}{4}$ to an improper fraction:

$3\frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$

Convert $2\frac{1}{2}$ to an improper fraction:

$2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$

Now subtract the improper fractions: $\frac{15}{4} - \frac{5}{2}$.

To subtract fractions with different denominators, find a common denominator. The denominators are 4 and 2. The least common multiple (LCM) of 4 and 2 is 4.

Convert $\frac{5}{2}$ to an equivalent fraction with denominator 4 by multiplying the numerator and denominator by $\frac{4}{2} = 2$:

$\frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4}$

Now subtract the fractions with the common denominator:

Difference in weight = $\frac{15}{4} - \frac{10}{4} = \frac{15 - 10}{4} = \frac{5}{4}$

The result is the improper fraction $\frac{5}{4}$ kg. We can convert this to a mixed fraction by dividing 5 by 4.

$5 \div 4$. The quotient is 1 and the remainder is $5 - (4 \times 1) = 5 - 4 = 1$.

So, $\frac{5}{4} = 1\frac{1}{4}$.

Alternate Solution:

Method 2: Subtract whole numbers and fractions separately.

The subtraction is $3\frac{3}{4} - 2\frac{1}{2}$.

We can write $3\frac{3}{4}$ as $3 + \frac{3}{4}$ and $2\frac{1}{2}$ as $2 + \frac{1}{2}$.

The expression is $(3 + \frac{3}{4}) - (2 + \frac{1}{2})$.

This is equal to $(3 - 2) + (\frac{3}{4} - \frac{1}{2})$.

Subtract the whole numbers: $3 - 2 = 1$.

Subtract the fractional parts: $\frac{3}{4} - \frac{1}{2}$. Find a common denominator (LCM of 4 and 2 is 4).

Convert $\frac{1}{2}$ to an equivalent fraction with denominator 4: $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

Now subtract the fractions: $\frac{3}{4} - \frac{2}{4} = \frac{3 - 2}{4} = \frac{1}{4}$.

Combine the difference of the whole numbers and the difference of the fractional parts:

Result $= 1 + \frac{1}{4} = 1\frac{1}{4}$.

Both methods yield the same result.

Final Answer:

Neetu’s fish weighed $\frac{5}{4}$ kg or $1\frac{1}{4}$ kg more than that of Narendra.

Given:

Length of cloth needed for the skirt = $1\frac{3}{4}$ m

Length of cloth needed for the scarf = $\frac{1}{2}$ m

To Find:

The total length of cloth Neelam's father must buy.

Solution:

To find the total length of cloth needed, we add the length of cloth required for the skirt and the scarf.

Total cloth needed = Cloth for skirt + Cloth for scarf

Total cloth needed = $1\frac{3}{4} + \frac{1}{2}$ m

Method 1: Convert the mixed fraction to an improper fraction and then add.

Convert $1\frac{3}{4}$ to an improper fraction:

$1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}$

Now add the improper fraction and the given fraction: $\frac{7}{4} + \frac{1}{2}$.

To add fractions with different denominators, we find a common denominator. The denominators are 4 and 2. The least common multiple (LCM) of 4 and 2 is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 4 by multiplying the numerator and denominator by $\frac{4}{2} = 2$:

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

Now add the fractions with the common denominator:

Total cloth needed = $\frac{7}{4} + \frac{2}{4} = \frac{7 + 2}{4} = \frac{9}{4}$

The result is the improper fraction $\frac{9}{4}$ m. We can convert this to a mixed fraction by dividing 9 by 4.

$9 \div 4$. The quotient is 2 and the remainder is $9 - (4 \times 2) = 9 - 8 = 1$.

So, $\frac{9}{4} = 2\frac{1}{4}$.

Alternate Solution:

Method 2: Add the whole number and the fractional parts separately.

The sum is $1\frac{3}{4} + \frac{1}{2} = (1 + \frac{3}{4}) + \frac{1}{2}$.

Group the whole number and the fractional parts:

$= 1 + (\frac{3}{4} + \frac{1}{2})$

Add the fractional parts: $\frac{3}{4} + \frac{1}{2}$. The common denominator for 4 and 2 is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with denominator 4: $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

Add the fractions: $\frac{3}{4} + \frac{2}{4} = \frac{3 + 2}{4} = \frac{5}{4}$.

The sum of the fractional parts is $\frac{5}{4}$. This is an improper fraction, which can be written as a mixed number $1\frac{1}{4}$.

Combine the whole number part (1) with the sum of the fractional parts ($1\frac{1}{4}$):

Result $= 1 + 1\frac{1}{4} = 1 + 1 + \frac{1}{4} = 2 + \frac{1}{4} = 2\frac{1}{4}$.

Both methods yield the same result.

Final Answer:

Neelam’s father must buy a total of $\frac{9}{4}$ m or $2\frac{1}{4}$ m of cloth.

Given:

Two additions of fractions are provided, showing the steps and the result.

To Analyze:

Identify the error in each provided addition.

Solution:

Let's analyze each addition separately.

(a) Addition of $8\frac{1}{2}$ and $4\frac{1}{4}$:

The calculation shown is: $8\frac{1}{2} + 4\frac{1}{4}$.

First step shows converting $8\frac{1}{2}$ to $8\frac{2}{4}$ and $4\frac{1}{4}$ as $4\frac{1}{4}$. This step is correct as it finds a common denominator (4) for the fractional parts.

$8\frac{1}{2} = 8 + \frac{1}{2} = 8 + \frac{1 \times 2}{2 \times 2} = 8 + \frac{2}{4} = 8\frac{2}{4}$

$4\frac{1}{4} = 4 + \frac{1}{4}$

The addition is $(8 + \frac{2}{4}) + (4 + \frac{1}{4})$.

Adding the whole number parts: $8 + 4 = 12$.

Adding the fractional parts: $\frac{2}{4} + \frac{1}{4} = \frac{2 + 1}{4} = \frac{3}{4}$.

Combining the results: $12 + \frac{3}{4} = 12\frac{3}{4}$.

The provided result is $12\frac{3}{8}$. The error occurred in the addition of the fractional parts. Instead of keeping the common denominator (4), the denominators were added ($4+4=8$), which is incorrect.

Error in (a): The denominators of the fractional parts were added instead of keeping the common denominator.

(b) Addition of $6\frac{1}{2}$ and $2\frac{1}{4}$:

The calculation shown is: $6\frac{1}{2} + 2\frac{1}{4}$.

The result is given as $8\frac{2}{6}$, which is simplified to $8\frac{1}{3}$.

Let's perform the addition correctly.

We add the whole number parts and the fractional parts separately: $(6 + \frac{1}{2}) + (2 + \frac{1}{4}) = (6 + 2) + (\frac{1}{2} + \frac{1}{4})$.

Whole number sum: $6 + 2 = 8$.

Fractional part sum: $\frac{1}{2} + \frac{1}{4}$. To add these, find a common denominator. The LCM of 2 and 4 is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with denominator 4: $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

Now add the fractional parts: $\frac{2}{4} + \frac{1}{4} = \frac{2 + 1}{4} = \frac{3}{4}$.

Combine the results: $8 + \frac{3}{4} = 8\frac{3}{4}$.

The provided result $8\frac{2}{6}$ was obtained by adding the whole numbers ($6+2=8$) and adding the numerators ($1+1=2$) and adding the denominators ($2+4=6$) of the original fractions directly, forming $8\frac{2}{6}$. Adding denominators in this way is incorrect.

Error in (b): The numerators were added together and the denominators were added together, without finding a common denominator first.

Summary of Errors:

In (a), the error is adding the denominators of the equivalent fractions.

In (b), the error is adding the numerators and the denominators of the original fractions directly.

Given:

We need to compare two quantities:

Quantity 1: 1 metre 40 centimetres + 60 centimetres

Quantity 2: 2.6 metres

To Find:

Which of the two given quantities is greater.

Solution:

To compare the two quantities, we need to express them in the same unit. We can convert everything to metres or to centimetres.

Let's convert everything to metres. We know that 1 metre = 100 centimetres, so 1 centimetre = $\frac{1}{100}$ metres or 0.01 metres.

Calculate Quantity 1:

1 metre 40 centimetres = 1 metre + 40 centimetres

40 centimetres = $40 \times 0.01$ metres = 0.40 metres.

So, 1 metre 40 centimetres = $1 \text{ m} + 0.40 \text{ m} = 1.40 \text{ m}$.

The quantity is $1.40 \text{ m} + 60 \text{ centimetres}$.

Convert 60 centimetres to metres:

60 centimetres = $60 \times 0.01$ metres = 0.60 metres.

Quantity 1 = $1.40 \text{ m} + 0.60 \text{ m}$

Add the decimal numbers:

So, Quantity 1 = 2.00 metres or 2 metres.

Calculate Quantity 2:

Quantity 2 is given as 2.6 metres.

Now, compare Quantity 1 (2.0 metres) and Quantity 2 (2.6 metres).

Comparing 2.0 and 2.6:

The whole number parts are 2 for both numbers.

Compare the tenths digits: In 2.0, the tenths digit is 0. In 2.6, the tenths digit is 6.

Since 6 is greater than 0, 2.6 is greater than 2.0.

$2.6 > 2.0$

Therefore, 2.6 metres is greater than 1 metre 40 centimetres + 60 centimetres.

Final Answer:

2.6 metres is greater than 1 metre 40 centimetres + 60 centimetres.

Given:

Fractions in Column I and figures representing fractions in Column II.

To Match:

Match each fraction in Column I with the corresponding figure in Column II.

Solution:

We will determine the fraction represented by each figure in Column II.

Figure (A): This figure is divided into 4 equal parts, and 3 parts are shaded. It represents the fraction $\frac{3}{4}$.

Figure (B): This figure is divided into 10 equal parts, and 6 parts are shaded. It represents the fraction $\frac{6}{10}$.

Figure (C): This figure is divided into 6 equal parts, and all 6 parts are shaded. It represents the fraction $\frac{6}{6}$ or 1.

Figure (D): This figure is divided into 16 equal parts, and 6 parts are marked with dots. It represents the fraction $\frac{6}{16}$.

Figure (E): This consists of two identical figures, each divided into 4 equal parts. The first figure has all 4 parts shaded (representing 1 whole), and the second figure has 2 parts shaded (representing $\frac{2}{4}$). The total shaded parts represent $1 + \frac{2}{4} = \frac{4}{4} + \frac{2}{4} = \frac{6}{4}$.

Now, we match the fractions from Column I with the fractions represented by the figures in Column II:

(i) $\frac{6}{4}$ matches with Figure (E).

(ii) $\frac{6}{10}$ matches with Figure (B).

(iii) $\frac{6}{6}$ matches with Figure (C).

(iv) $\frac{6}{16}$ matches with Figure (D).

Figure (A) representing $\frac{3}{4}$ does not have a match in Column I.

Final Answer:

The correct matches are:

(i) $\to$ (E)

(ii) $\to$ (B)

(iii) $\to$ (C)

(iv) $\to$ (D)

Given:

The collection of numbers is {0, 1, 2, 3, 4, 5}.

To Find:

1. The fraction of natural numbers to the total number of numbers in the collection.

2. The fraction of whole numbers to the total number of numbers in the collection.

Solution:

First, let's determine the total number of numbers in the given collection {0, 1, 2, 3, 4, 5}.

Counting the numbers, we find there are 6 numbers in the collection.

Total number of numbers = 6.

Next, let's identify the natural numbers in this collection.

Natural numbers are the counting numbers, starting from 1 (i.e., {1, 2, 3, 4, 5, ...}).

In the given collection {0, 1, 2, 3, 4, 5}, the natural numbers are 1, 2, 3, 4, and 5.

Number of natural numbers in the collection = 5.

The fraction of natural numbers to the total number of numbers is:

Fraction (Natural Numbers) = $\frac{\text{Number of natural numbers}}{\text{Total number of numbers}} = \frac{5}{6}$

Now, let's identify the whole numbers in this collection.

Whole numbers are natural numbers including zero (i.e., {0, 1, 2, 3, 4, 5, ...}).

In the given collection {0, 1, 2, 3, 4, 5}, all the numbers are whole numbers: 0, 1, 2, 3, 4, and 5.

Number of whole numbers in the collection = 6.

The fraction of whole numbers to the total number of numbers is:

Fraction (Whole Numbers) = $\frac{\text{Number of whole numbers}}{\text{Total number of numbers}} = \frac{6}{6}$

The fraction $\frac{6}{6}$ simplifies to 1.

Final Answer:

The fraction that represents the number of natural numbers to the total numbers in the collection is $\frac{5}{6}$.

The fraction that represents the number of whole numbers to the total numbers in the collection is $\frac{6}{6}$ or 1.

Given:

The collection of numbers is {-3, -2, -1, 0, 1, 2, 3}.

To Find:

1. The fraction of natural numbers to the total numbers in the collection.

2. The fraction of whole numbers to the total numbers in the collection.

3. The fraction of integers to the total numbers in the collection.

Solution:

First, let's determine the total number of numbers in the given collection {-3, -2, -1, 0, 1, 2, 3}.

Counting the numbers, we find there are 7 numbers in the collection.

Total number of numbers = 7.

Fraction for Natural Numbers:

Natural numbers are the counting numbers, starting from 1 (i.e., {1, 2, 3, 4, 5, ...}).

In the given collection {-3, -2, -1, 0, 1, 2, 3}, the natural numbers are 1, 2, and 3.

Number of natural numbers in the collection = 3.

The fraction of natural numbers to the total number of numbers is:

Fraction (Natural Numbers) = $\frac{\text{Number of natural numbers}}{\text{Total number of numbers}} = \frac{3}{7}$

This fraction cannot be simplified further as 3 and 7 are relatively prime.

Fraction for Whole Numbers:

Whole numbers are natural numbers including zero (i.e., {0, 1, 2, 3, 4, 5, ...}).

In the given collection {-3, -2, -1, 0, 1, 2, 3}, the whole numbers are 0, 1, 2, and 3.

Number of whole numbers in the collection = 4.

The fraction of whole numbers to the total number of numbers is:

Fraction (Whole Numbers) = $\frac{\text{Number of whole numbers}}{\text{Total number of numbers}} = \frac{4}{7}$

This fraction cannot be simplified further as 4 and 7 are relatively prime.

Fraction for Integers:

Integers are all whole numbers and their negative counterparts (i.e., {..., -3, -2, -1, 0, 1, 2, 3, ...}).

In the given collection {-3, -2, -1, 0, 1, 2, 3}, all the numbers are integers.

Number of integers in the collection = 7.

The fraction of integers to the total number of numbers is:

Fraction (Integers) = $\frac{\text{Number of integers}}{\text{Total number of numbers}} = \frac{7}{7}$

This fraction simplifies to 1.

Final Answer:

The fraction for natural numbers is $\frac{3}{7}$.

The fraction for whole numbers is $\frac{4}{7}$.

The fraction for integers is $\frac{7}{7}$ or 1.

Given:

Sum of two fractions is $\frac{7}{11}$.

Difference of the same two fractions is $\frac{2}{11}$.

To Find:

A pair of fractions that satisfy the given conditions.

Solution:

Let the two fractions be $x$ and $y$. We can set up a system of two linear equations based on the given information.

Sum: $x + y = \frac{7}{11}$

Difference: Let's assume $x \ge y$. Then the difference is $x - y = \frac{2}{11}$.

We can solve this system of equations for $x$ and $y$. Add equation (i) and equation (ii).

$(x + y) + (x - y) = \frac{7}{11} + \frac{2}{11}$

$x + y + x - y = \frac{7 + 2}{11}$

$2x = \frac{9}{11}$

Now, solve for $x$ by multiplying both sides by $\frac{1}{2}$ (or dividing by 2).

$x = \frac{9}{11} \times \frac{1}{2}$

$x = \frac{9 \times 1}{11 \times 2}$

$x = \frac{9}{22}$

Now that we have the value of $x$, we can substitute it back into either equation (i) or (ii) to find the value of $y$. Let's use equation (i).

$x + y = \frac{7}{11}$

$\frac{9}{22} + y = \frac{7}{11}$

Subtract $\frac{9}{22}$ from both sides to find $y$.

$y = \frac{7}{11} - \frac{9}{22}$

To subtract the fractions, find a common denominator. The denominators are 11 and 22. The LCM of 11 and 22 is 22.

Convert $\frac{7}{11}$ to an equivalent fraction with denominator 22 by multiplying the numerator and denominator by $\frac{22}{11} = 2$:

$\frac{7}{11} = \frac{7 \times 2}{11 \times 2} = \frac{14}{22}$

Now subtract the fractions:

$y = \frac{14}{22} - \frac{9}{22} = \frac{14 - 9}{22} = \frac{5}{22}$

So, the two fractions are $\frac{9}{22}$ and $\frac{5}{22}$.

Let's verify the sum and difference.

Sum = $\frac{9}{22} + \frac{5}{22} = \frac{9+5}{22} = \frac{14}{22}$. This simplifies to $\frac{14 \div 2}{22 \div 2} = \frac{7}{11}$. This matches the given sum.

Difference = $\frac{9}{22} - \frac{5}{22} = \frac{9-5}{22} = \frac{4}{22}$. This simplifies to $\frac{4 \div 2}{22 \div 2} = \frac{2}{11}$. This matches the given difference.

The pair of fractions is $\frac{9}{22}$ and $\frac{5}{22}$.

Final Answer:

A pair of fractions whose sum is $\frac{7}{11}$ and difference is $\frac{2}{11}$ is $\frac{9}{22}$ and $\frac{5}{22}$.

Given:

We are asked to find the fraction of a straight angle that is a right angle.

To Find:

The fraction $\frac{\text{Right Angle}}{\text{Straight Angle}}$.

Solution:

We know the measures of a straight angle and a right angle in degrees.

A straight angle measures $180^\circ$.

A right angle measures $90^\circ$.

The fraction of a straight angle that is a right angle is given by the ratio of the measure of a right angle to the measure of a straight angle.

Fraction = $\frac{\text{Measure of Right Angle}}{\text{Measure of Straight Angle}}$

Fraction = $\frac{90^\circ}{180^\circ} = \frac{90}{180}$

Now, we simplify the fraction $\frac{90}{180}$. We can divide both the numerator and the denominator by their greatest common divisor, which is 90.

$\frac{90}{180} = \frac{90 \div 90}{180 \div 90} = \frac{1}{2}$

So, a right angle is $\frac{1}{2}$ of a straight angle.

Final Answer:

A right angle is $\frac{1}{2}$ fraction of a straight angle.

Given:

A list of fractions (cards) and three categories representing fractions less than 1, equal to 1, and greater than 1 (bags).

To Put:

Place each fraction card into the correct bag.

Solution:

To determine which bag a fraction belongs to, we compare its value to 1. This can be done by comparing the numerator and the denominator:

• If the numerator is less than the denominator, the fraction is less than 1.
• If the numerator is equal to the denominator, the fraction is equal to 1.
• If the numerator is greater than the denominator, the fraction is greater than 1.

Let's analyze each fraction card:

(i) $\frac{3}{7}$: Numerator (3) < Denominator (7). Value is less than 1.

(ii) $\frac{4}{4}$: Numerator (4) = Denominator (4). Value is equal to 1.

(iii) $\frac{9}{8}$: Numerator (9) > Denominator (8). Value is greater than 1.

(iv) $\frac{8}{9}$: Numerator (8) < Denominator (9). Value is less than 1.

(v) $\frac{5}{6}$: Numerator (5) < Denominator (6). Value is less than 1.

(vi) $\frac{6}{11}$: Numerator (6) < Denominator (11). Value is less than 1.

(vii) $\frac{18}{18}$: Numerator (18) = Denominator (18). Value is equal to 1.

(viii) $\frac{19}{25}$: Numerator (19) < Denominator (25). Value is less than 1.

(ix) $\frac{2}{3}$: Numerator (2) < Denominator (3). Value is less than 1.

(x) $\frac{13}{17}$: Numerator (13) < Denominator (17). Value is less than 1.

Now, we can group the fractions based on their values relative to 1.

Final Answer:

The fractions are placed in the correct bags as follows:

Bag 1 (< 1): These are proper fractions.

$\frac{3}{7}, \frac{8}{9}, \frac{5}{6}, \frac{6}{11}, \frac{19}{25}, \frac{2}{3}, \frac{13}{17}$

Bag 2 (= 1): These are fractions where the numerator equals the denominator.

$\frac{4}{4}, \frac{18}{18}$

Bag 3 (> 1): These are improper fractions where the numerator is greater than the denominator.

$\frac{9}{8}$