Chapter 2 Fractions and Decimals (Additional Questions)

A proper fraction is a fraction where the numerator is less than the denominator.

Let's examine each option:

(A) $\frac{7}{5}$: The numerator (7) is greater than the denominator (5). This is an improper fraction.

(B) $\frac{12}{13}$: The numerator (12) is less than the denominator (13). This is a proper fraction.

(C) $\frac{9}{9}$: The numerator (9) is equal to the denominator (9). This is an improper fraction.

(D) $1\frac{1}{2}$: This is a mixed number, which can be written as $\frac{3}{2}$. The numerator (3) is greater than the denominator (2). This is an improper fraction.

Therefore, the only proper fraction among the given options is $\frac{12}{13}$.

The correct option is (B).

To convert a mixed fraction $a\frac{b}{c}$ into an improper fraction, we use the formula:

Improper fraction = $\frac{(a \times c) + b}{c}$

Given the mixed fraction $3\frac{2}{5}$, here $a=3$, $b=2$, and $c=5$.

Applying the formula:

Numerator = (Whole number $\times$ Denominator) + Numerator of fractional part

Numerator = $(3 \times 5) + 2 = 15 + 2 = 17$

The denominator remains the same as the original fraction, which is 5.

So, the improper fraction is $\frac{17}{5}$.

Comparing this with the given options, we find that option (C) matches our result.

The correct option is (C).

To add fractions with different denominators, we first need to find a common denominator.

The given fractions are $\frac{1}{3}$ and $\frac{1}{6}$.

The denominators are 3 and 6.

The least common multiple (LCM) of 3 and 6 is 6.

Now, we convert each fraction to an equivalent fraction with a denominator of 6.

For $\frac{1}{3}$: We need to multiply the denominator 3 by 2 to get 6. We must do the same to the numerator.

$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

For $\frac{1}{6}$: The denominator is already 6, so it remains the same.

$\frac{1}{6}$

Now, we add the equivalent fractions which have the same denominator:

$\frac{2}{6} + \frac{1}{6} = \frac{2+1}{6} = \frac{3}{6}$

The resulting fraction $\frac{3}{6}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

$\frac{\cancel{3}^{1}}{\cancel{6}_{2}} = \frac{1}{2}$

So, the sum of $\frac{1}{3} + \frac{1}{6}$ is $\frac{1}{2}$.

Comparing this result with the given options, we find that option (D) matches our result.

The correct option is (D).

To subtract mixed fractions, we can first convert them into improper fractions.

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

$2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$

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 we need to subtract $\frac{5}{4}$ from $\frac{5}{2}$.

The expression is $\frac{5}{2} - \frac{5}{4}$.

To subtract these fractions, we need a common denominator.

The denominators are 2 and 4.

The least common multiple (LCM) of 2 and 4 is 4.

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

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

Now perform the subtraction:

$\frac{10}{4} - \frac{5}{4} = \frac{10 - 5}{4} = \frac{5}{4}$

The result is the improper fraction $\frac{5}{4}$. Let's convert this back to a mixed fraction to compare with the options.

Divide 5 by 4:

$5 \div 4 = 1$ with a remainder of $5 - (1 \times 4) = 1$.

So, $\frac{5}{4} = 1\frac{1}{4}$.

Comparing the result $1\frac{1}{4}$ with the given options, we find that option (B) matches our result.

The correct option is (B).

To multiply an integer by a fraction, we can write the integer as a fraction with a denominator of 1, and then multiply the numerators together and the denominators together.

The expression is $8 \times \frac{3}{4}$.

We can write 8 as $\frac{8}{1}$.

So, the expression becomes $\frac{8}{1} \times \frac{3}{4}$.

Multiply the numerators: $8 \times 3 = 24$.

Multiply the denominators: $1 \times 4 = 4$.

The product is $\frac{24}{4}$.

Now, simplify the fraction $\frac{24}{4}$:

$\frac{24}{4} = 24 \div 4 = 6$.

Alternatively, we can simplify before multiplying.

$8 \times \frac{3}{4} = \frac{8 \times 3}{4}$

We can cancel out a common factor of 4 from the numerator (8) and the denominator (4).

$\frac{\cancel{8}^{2} \times 3}{\cancel{4}_{1}} = \frac{2 \times 3}{1} = \frac{6}{1} = 6$.

The product is 6.

Comparing this result with the given options, we find that option (A) matches our result.

The correct option is (A).

The reciprocal of a fraction is found by swapping the numerator and the denominator. If the fraction is $\frac{a}{b}$, its reciprocal is $\frac{b}{a}$.

Given the fraction $\frac{5}{7}$.

Here, the numerator is 5 and the denominator is 7.

To find the reciprocal, we swap the numerator and the denominator.

The new numerator will be 7, and the new denominator will be 5.

So, the reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$.

We can verify this because the product of a fraction and its reciprocal is always 1:

$\frac{5}{7} \times \frac{7}{5} = \frac{5 \times 7}{7 \times 5} = \frac{35}{35} = 1$

Comparing our result $\frac{7}{5}$ with the given options, we find that option (B) matches our result.

The correct option is (B).

To divide a number by a fraction, we multiply the number by the reciprocal of the fraction.

The given division is $6 \div \frac{1}{3}$.

First, find the reciprocal of the fraction $\frac{1}{3}$.

The reciprocal of $\frac{1}{3}$ is obtained by swapping the numerator and the denominator, which is $\frac{3}{1}$ or simply 3.

Now, replace the division by multiplication and use the reciprocal:

$6 \div \frac{1}{3} = 6 \times 3$

Perform the multiplication:

$6 \times 3 = 18$

So, $6 \div \frac{1}{3} = 18$.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

In a decimal number, the place values of the digits to the right of the decimal point are, from left to right: tenths, hundredths, thousandths, and so on.

The given decimal number is $52.36$.

Let's look at the digits and their positions relative to the decimal point:

• The digit 5 is to the left of the decimal point, in the tens place.
• The digit 2 is to the left of the decimal point, in the units (or ones) place.
• The digit 3 is the first digit immediately to the right of the decimal point.
• The digit 6 is the second digit to the right of the decimal point.

The place value of the first digit to the right of the decimal point is the tenths place.

The place value of the second digit to the right of the decimal point is the hundredths place.

Since the digit 3 is the first digit to the right of the decimal point in $52.36$, its place value is tenths.

Comparing our result with the given options, we find that option (A) matches our result.

The correct option is (A).

To find the smallest decimal among the given options, we compare them digit by digit, starting from the leftmost digit after the decimal point (the tenths place).

The given decimal numbers are:

• (A) 0.125
• (B) 0.215
• (C) 0.152
• (D) 0.251

Let's compare the digits in the tenths place:

• 0.125 has 1 in the tenths place.
• 0.215 has 2 in the tenths place.
• 0.152 has 1 in the tenths place.
• 0.251 has 2 in the tenths place.

Decimals with a smaller tenths digit are smaller. So, 0.125 and 0.152 are smaller than 0.215 and 0.251.

Now, let's compare 0.125 and 0.152. We compare the digits in the hundredths place:

• 0.125 has 2 in the hundredths place.
• 0.152 has 5 in the hundredths place.

Since 2 is less than 5, 0.125 is smaller than 0.152.

Comparing 0.125 with the other options (0.215 and 0.251, which have larger tenths digits), 0.125 is the smallest.

Therefore, the smallest decimal is 0.125.

The correct option is (A).

To add decimal numbers, we align the decimal points vertically and then add the numbers just like whole numbers. We can add trailing zeros to make the number of decimal places equal.

The numbers are $3.5$ and $2.05$.

$3.5$ can be written as $3.50$ to have two decimal places, matching $2.05$.

Now, we add $3.50$ and $2.05$:

$3.50$

$+$ $2.05$

$------$

$5.55$

Adding the digits in the hundredths place: $0 + 5 = 5$.

Adding the digits in the tenths place: $5 + 0 = 5$.

Place the decimal point.

Adding the digits in the units place: $3 + 2 = 5$.

The sum is $5.55$.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

To multiply a decimal number by 10, 100, 1000, etc., we shift the decimal point to the right by as many places as there are zeros in the multiplier.

The given multiplication is $4.5 \times 100$.

The multiplier is 100, which has two zeros.

So, we need to shift the decimal point in $4.5$ two places to the right.

$4.5$

Shifting the decimal point one place to the right gives $45$.

To shift it a second place, we need to add a zero after 5.

$45.0$

Shifting the decimal point a second place to the right gives $450$.

Thus, $4.5 \times 100 = 450$.

Comparing this result with the given options, we find that option (C) matches our result.

The correct option is (C).

To divide a decimal number by a whole number, we perform long division as usual, placing the decimal point in the quotient directly above the decimal point in the dividend.

We need to divide $12.6$ by $3$.

Let's perform the division:

Divide the whole part first: $12 \div 3 = 4$. Write 4 in the quotient above 2.

Place the decimal point in the quotient directly above the decimal point in 12.6.

Bring down the next digit, which is 6.

Now, divide 6 by 3: $6 \div 3 = 2$. Write 2 in the quotient above 6.

The division can be shown as:

$\begin{array}{r} 4.2 \\ 3{\overline{\smash{\big)}\,12.6}} \\ \underline{-~\phantom{(}12\phantom{.0)}} \\ 0.6 \\ \underline{-~\phantom{(0.}6)} \\ 0.0 \end{array}$

So, $12.6 \div 3 = 4.2$.

Comparing this result with the given options, we find that option (A) matches our result.

The correct option is (A).

To evaluate the expression $\frac{1}{2} + \frac{1}{4} \times \frac{1}{2}$, we need to follow the order of operations (BODMAS/PEMDAS), which states that multiplication should be performed before addition.

First, calculate the product of $\frac{1}{4}$ and $\frac{1}{2}$:

$\frac{1}{4} \times \frac{1}{2} = \frac{1 \times 1}{4 \times 2} = \frac{1}{8}$

Now, substitute this result back into the original expression and perform the addition:

$\frac{1}{2} + \frac{1}{8}$

To add fractions with different denominators, we find a common denominator. The denominators are 2 and 8.

The least common multiple (LCM) of 2 and 8 is 8.

Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 8:

$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$

Now add the fractions with the same denominator:

$\frac{4}{8} + \frac{1}{8} = \frac{4+1}{8} = \frac{5}{8}$

The value of the expression $\frac{1}{2} + \frac{1}{4} \times \frac{1}{2}$ is $\frac{5}{8}$.

Comparing this result with the given options, we find that option (C) matches our result.

The correct option is (C).

A proper fraction is a fraction where the numerator is less than the denominator ($Numerator < Denominator$).

An improper fraction is a fraction where the numerator is greater than or equal to the denominator ($Numerator \ge Denominator$).

We are looking for the fraction that is NOT a proper fraction, which means we are looking for an improper fraction.

Let's examine each option:

• (A) $\frac{3}{7}$: The numerator is 3 and the denominator is 7. Since $3 < 7$, this is a proper fraction.
• (B) $\frac{9}{10}$: The numerator is 9 and the denominator is 10. Since $9 < 10$, this is a proper fraction.
• (C) $\frac{11}{8}$: The numerator is 11 and the denominator is 8. Since $11 > 8$, this is an improper fraction.
• (D) $\frac{1}{2}$: The numerator is 1 and the denominator is 2. Since $1 < 2$, this is a proper fraction.

The fraction that is NOT a proper fraction is $\frac{11}{8}$.

The correct option is (C).

Equivalent fractions represent the same part of a whole. They are obtained by multiplying or dividing both the numerator and the denominator of a fraction by the same non-zero number.

We need to check if each given option is equivalent to $\frac{2}{5}$.

Let's examine option (A) $\frac{4}{10}$.

We can check if we can multiply the numerator and denominator of $\frac{2}{5}$ by the same number to get $\frac{4}{10}$.

Numerator: $2 \times 2 = 4$

Denominator: $5 \times 2 = 10$

Since we multiplied both by 2, $\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$.

So, $\frac{4}{10}$ is equivalent to $\frac{2}{5}$.

Let's examine option (B) $\frac{6}{15}$.

We check if we can multiply the numerator and denominator of $\frac{2}{5}$ by the same number to get $\frac{6}{15}$.

Numerator: $2 \times 3 = 6$

Denominator: $5 \times 3 = 15$

Since we multiplied both by 3, $\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}$.

So, $\frac{6}{15}$ is equivalent to $\frac{2}{5}$.

Let's examine option (C) $\frac{10}{25}$.

We check if we can multiply the numerator and denominator of $\frac{2}{5}$ by the same number to get $\frac{10}{25}$.

Numerator: $2 \times 5 = 10$

Denominator: $5 \times 5 = 25$

Since we multiplied both by 5, $\frac{2}{5} = \frac{2 \times 5}{5 \times 5} = \frac{10}{25}$.

So, $\frac{10}{25}$ is equivalent to $\frac{2}{5}$.

Since options (A), (B), and (C) are all equivalent to $\frac{2}{5}$, the correct answer is (D).

The correct option is (D).

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

The given fraction is $\frac{3}{5}$.

We need to divide 3 by 5.

Performing the division:

$3 \div 5$

Since 3 is less than 5, we can write 3 as 3.0 and perform the division.

$\begin{array}{r} 0.6 \\ 5{\overline{\smash{\big)}\,3.0}} \\ \underline{-~\phantom{(}30\phantom{)}} \\ 0\phantom{0} \end{array}$

So, the decimal representation of $\frac{3}{5}$ is $0.6$.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

To match the fractions with their decimal equivalents, we convert each fraction to a decimal by dividing the numerator by the denominator.

(i) $\frac{1}{4}$:

$1 \div 4 = 0.25$

So, (i) corresponds to (d).

(ii) $\frac{1}{2}$:

$1 \div 2 = 0.5$

So, (ii) corresponds to (c).

(iii) $\frac{3}{4}$:

$3 \div 4 = 0.75$

So, (iii) corresponds to (b).

(iv) $\frac{1}{5}$:

$1 \div 5 = 0.2$

So, (iv) corresponds to (a).

The correct matching is:

• (i) - (d)
• (ii) - (c)
• (iii) - (b)
• (iv) - (a)

Comparing this matching with the given options, we find that option (B) matches our result.

The correct option is (B).

Let's evaluate the Assertion (A) and the Reason (R).

Assertion (A): The product of $\frac{2}{3}$ and $\frac{4}{5}$ is less than $\frac{2}{3}$.

Calculate the product:

Product = $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$

Now, compare the product ($\frac{8}{15}$) with the first fraction ($\frac{2}{3}$).

To compare $\frac{8}{15}$ and $\frac{2}{3}$, we find a common denominator, which is the LCM of 15 and 3. The LCM is 15.

Convert $\frac{2}{3}$ to an equivalent fraction with denominator 15:

$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$

Now compare $\frac{8}{15}$ and $\frac{10}{15}$.

Since the denominators are the same, we compare the numerators: $8 < 10$.

Therefore, $\frac{8}{15} < \frac{10}{15}$.

This means $\frac{2}{3} \times \frac{4}{5} < \frac{2}{3}$.

The Assertion (A) is true.

Reason (R): When a proper fraction is multiplied by another proper fraction, the product is smaller than the first fraction.

A proper fraction is a fraction where the numerator is less than the denominator. Both $\frac{2}{3}$ and $\frac{4}{5}$ are proper fractions.

Let the first proper fraction be $\frac{a}{b}$ and the second proper fraction be $\frac{c}{d}$, where $0 < a < b$ and $0 < c < d$.

The product is $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$.

We need to compare $\frac{ac}{bd}$ with $\frac{a}{b}$.

Since $\frac{c}{d}$ is a proper fraction, we know that $0 < \frac{c}{d} < 1$.

When we multiply a positive number (like $\frac{a}{b}$) by a number between 0 and 1 (like $\frac{c}{d}$), the result is smaller than the original number.

Mathematically, comparing $\frac{ac}{bd}$ and $\frac{a}{b}$ is equivalent to comparing $ac \times b$ and $a \times bd$. This simplifies to comparing $abc$ and $abd$. Since $a, b, d$ are positive, we can compare $c$ and $d$. From the definition of a proper fraction $\frac{c}{d}$, we know that $c < d$. Therefore, $abc < abd$, which means $\frac{ac}{bd} < \frac{a}{b}$.

The Reason (R) is true.

Relationship between A and R:

The Reason (R) provides a general rule about the product of two proper fractions. The Assertion (A) is a specific example where this rule applies (multiplying the proper fraction $\frac{2}{3}$ by the proper fraction $\frac{4}{5}$ results in a product smaller than $\frac{2}{3}$). Thus, Reason (R) correctly explains Assertion (A).

Both A and R are true, and R is the correct explanation of A.

The correct option is (A).

Let's evaluate the Assertion (A) and the Reason (R).

Assertion (A): Dividing $5.2$ by $0.1$ gives $52$.

To perform the division $5.2 \div 0.1$, we can multiply both the dividend and the divisor by 10 to eliminate the decimal in the divisor:

$5.2 \times 10 = 52$

$0.1 \times 10 = 1$

So, $5.2 \div 0.1$ is equivalent to $52 \div 1$.

$52 \div 1 = 52$

The result of dividing $5.2$ by $0.1$ is indeed $52$.

The Assertion (A) is true.

Reason (R): Dividing by a decimal less than 1 results in a quotient greater than the dividend.

Consider a positive dividend, say $D$, and a divisor, say $d$, where $0 < d < 1$.

We are looking at the value of $D \div d$.

Since $0 < d < 1$, we can express $d$ as a fraction $\frac{p}{q}$, where $p < q$. For example, $0.1 = \frac{1}{10}$, $0.5 = \frac{1}{2}$, $0.25 = \frac{1}{4}$.

Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $d = \frac{p}{q}$ is $\frac{q}{p}$.

So, $D \div d = D \div \frac{p}{q} = D \times \frac{q}{p}$.

Since $p < q$, the fraction $\frac{q}{p}$ is greater than 1 ($\frac{q}{p} > 1$).

When a positive number $D$ is multiplied by a number greater than 1 (which is $\frac{q}{p}$), the product ($D \times \frac{q}{p}$) is greater than the original number ($D$).

Therefore, the quotient $D \div d$ is greater than the dividend $D$.

The Reason (R) is true.

Relationship between A and R:

In Assertion (A), the dividend is $5.2$ and the divisor is $0.1$. The divisor $0.1$ is a decimal less than 1. The quotient is $52$, which is greater than the dividend $5.2$. This specific case demonstrates the principle stated in Reason (R).

Reason (R) provides the general mathematical principle that explains why the result of dividing $5.2$ by $0.1$ (a number less than 1) is larger than $5.2$.

Thus, Reason (R) is the correct explanation for Assertion (A).

Both A and R are true, and R is the correct explanation of A.

The correct option is (A).

First, we need to find the total cost of the items Reena bought.

Cost of geometry box = $\textsf{₹ }52.75$

Cost of notebook = $\textsf{₹ }15.50$

Total cost = Cost of geometry box + Cost of notebook

Total cost = $\textsf{₹ }52.75 + \textsf{₹ }15.50$

We can add the decimal numbers:

$52.75$

$+ \ 15.50$

$-------$

$7 \phantom{.} 8.25$

So, the total cost is $\textsf{₹ }68.25$.

Next, we need to calculate the change Reena received. The change is the amount paid minus the total cost.

Amount paid = $\textsf{₹ }100$

Total cost = $\textsf{₹ }68.25$

Change = Amount paid - Total cost

Change = $\textsf{₹ }100.00 - \textsf{₹ }68.25$ (Writing 100 as 100.00 for decimal subtraction)

We perform the subtraction:

$100.00$

$- \ \ \ \ 68.25$

$--------$

$\ \ \ 31.75$

So, the change Reena got is $\textsf{₹ }31.75$.

Comparing this result with the given options, we find that option (A) matches our result.

The correct option is (A).

To find the total length of cloth used, we need to add the length of cloth used for the shirt and the length of cloth used for the t-shirt.

Length of cloth for shirt = $\frac{3}{4}\text{ m}$

Length of cloth for t-shirt = $\frac{1}{2}\text{ m}$

Total length of cloth used = $\frac{3}{4} + \frac{1}{2}$

To add these fractions, we need to 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:

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

Now add the fractions with the same denominator:

$\frac{3}{4} + \frac{2}{4} = \frac{3+2}{4} = \frac{5}{4}$

The total length of cloth used is $\frac{5}{4}\text{ m}$.

The fraction $\frac{5}{4}$ is an improper fraction. We can convert it to a mixed number to compare with the options.

Divide 5 by 4: $5 \div 4 = 1$ with a remainder of $5 - (1 \times 4) = 1$.

So, $\frac{5}{4} = 1\frac{1}{4}$.

The total length of cloth used is $1\frac{1}{4}\text{ m}$.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

The distance covered by the car is proportional to the amount of petrol used.

Distance covered in 1 litre = $15.5\text{ km}$

Quantity of petrol = $3.5$ litres

Distance covered in $3.5$ litres = Distance in 1 litre $\times$ Quantity of petrol

Distance covered = $15.5 \times 3.5$

We need to multiply $15.5$ by $3.5$. We can ignore the decimal points and multiply 155 by 35, and then place the decimal point in the result.

$155 \times 35$

$\begin{array}{cc}& & 1 & 5 & 5 \\ \times & & & 3 & 5 \\ \hline && 7 & 7 & 5 \\ & 4 & 6 & 5 & \times \\ \hline 5 & 4 & 2 & 5 \\ \hline \end{array}$

The sum is 5425.

Now, count the total number of digits after the decimal points in the original numbers.

In $15.5$, there is 1 digit after the decimal point.

In $3.5$, there is 1 digit after the decimal point.

Total number of digits after the decimal points = $1 + 1 = 2$.

So, we place the decimal point two places from the right in the product 5425.

The result is $54.25$.

The car will cover $54.25\text{ km}$ in $3.5$ litres of petrol.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

To find the number of pieces of ribbon, we need to divide the total length of the ribbon by the length of each piece.

Total length of ribbon = $2\text{ m}$

Length of each piece = $\frac{1}{5}\text{ m}$

Number of pieces = Total length $\div$ Length of each piece

Number of pieces = $2 \div \frac{1}{5}$

To divide by a fraction, we multiply by its reciprocal.

The reciprocal of $\frac{1}{5}$ is $\frac{5}{1}$, which is 5.

Number of pieces = $2 \times 5$

Number of pieces = $10$

So, 10 pieces of ribbon, each $\frac{1}{5}\text{ m}$ long, can be cut from a $2\text{ m}$ long ribbon.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

To divide a number by a fraction, we multiply the number by the reciprocal of the fraction.

The expression is $10 \div \frac{1}{2}$.

First, find the reciprocal of the fraction $\frac{1}{2}$.

The reciprocal of $\frac{1}{2}$ is obtained by swapping the numerator and the denominator, which is $\frac{2}{1}$ or simply 2.

Now, replace the division by multiplication and use the reciprocal:

$10 \div \frac{1}{2} = 10 \times 2$

Perform the multiplication:

$10 \times 2 = 20$

So, $10 \div \frac{1}{2} = 20$.

The blank should be filled with 20.

Comparing this result with the given options, we find that option (D) matches our result.

The correct option is (D).

To multiply decimal numbers, we first multiply the numbers as if they were whole numbers. Then, we count the total number of digits after the decimal points in the original numbers and place the decimal point in the product such that there are that many digits after the decimal point.

The expression is $0.4 \times 0.6$.

First, multiply the numbers without the decimal points: $4 \times 6 = 24$.

Now, count the number of decimal places in each original number:

• $0.4$ has 1 digit after the decimal point.
• $0.6$ has 1 digit after the decimal point.

The total number of decimal places is $1 + 1 = 2$.

Place the decimal point in the product 24 such that there are 2 digits after the decimal point. Starting from the right, move the decimal point 2 places to the left.

The result is $0.24$.

So, $0.4 \times 0.6 = 0.24$.

The blank should be filled with 0.24.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

In tally marks, a group of five is represented by four vertical strokes crossed by a diagonal stroke ($\bcancel{||||}$). Numbers less than five are represented by the corresponding number of vertical strokes ($|$, $||$, $|||$, $||||$).

We need to represent the number 8 using tally marks.

We can group 8 into fives and the remaining count.

$8 = 5 + 3$

The number 5 is represented by $\bcancel{||||}$.

The number 3 is represented by $|||$.

So, the tally mark representation for 8 is a group of five followed by three vertical strokes: $\bcancel{||||}\ |||$.

Let's check the options:

• (A) $\bcancel{||||}\ ||$ represents $5 + 2 = 7$.
• (B) $\bcancel{||||}\ |||$ represents $5 + 3 = 8$.
• (C) $\bcancel{||||}$ represents 5.
• (D) $\bcancel{||||}\ ||||$ represents $5 + 4 = 9$.

The tally mark representation for the number 8 is $\bcancel{||||}\ |||$.

The correct option is (B).

To find the amount of juice left, we need to subtract the amount of juice consumed from the initial amount of juice in the container.

Initial amount of juice = $5\frac{1}{2}$ litres

Amount of juice consumed = $2\frac{3}{4}$ litres

Amount of juice left = Initial amount - Amount consumed

Amount of juice left = $5\frac{1}{2} - 2\frac{3}{4}$

First, convert the mixed fractions to improper fractions:

$5\frac{1}{2} = \frac{(5 \times 2) + 1}{2} = \frac{10 + 1}{2} = \frac{11}{2}$

$2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$

Now, perform the subtraction: $\frac{11}{2} - \frac{11}{4}$.

To subtract these fractions, we need 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 a denominator of 4:

$\frac{11}{2} = \frac{11 \times 2}{2 \times 2} = \frac{22}{4}$

Now subtract the fractions with the same denominator:

$\frac{22}{4} - \frac{11}{4} = \frac{22 - 11}{4} = \frac{11}{4}$

The amount of juice left is $\frac{11}{4}$ litres.

We can convert this improper fraction back to a mixed number to compare with the options.

Divide 11 by 4: $11 \div 4 = 2$ with a remainder of $11 - (2 \times 4) = 11 - 8 = 3$.

So, $\frac{11}{4} = 2\frac{3}{4}$.

The amount of juice left is $2\frac{3}{4}$ litres.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

To find the cost of $2.5\text{ kg}$ of apples, we need to multiply the cost of $1\text{ kg}$ by the quantity of apples.

Cost of $1\text{ kg}$ of apples = $\textsf{₹ }120.50$

Quantity of apples = $2.5\text{ kg}$

Total cost = Cost per kg $\times$ Quantity

Total cost = $\textsf{₹ }120.50 \times 2.5$

We need to multiply $120.50$ by $2.5$. We can ignore the decimal points and multiply 12050 by 25, and then place the decimal point in the result.

However, since $120.50$ is the same as $120.5$, we can multiply $120.5$ by $2.5$. We multiply 1205 by 25.

$1205 \times 25$

$\begin{array}{cc}& & 1 & 2 & 0 & 5 \\ \times & & & & 2 & 5 \\ \hline && 6 & 0 & 2 & 5 \\ & 2 & 4 & 1 & 0 & \times \\ \hline 3 & 0 & 1 & 2 & 5 \\ \hline \end{array}$

The product is 30125.

Now, count the total number of digits after the decimal points in the original numbers.

In $120.5$, there is 1 digit after the decimal point.

In $2.5$, there is 1 digit after the decimal point.

Total number of digits after the decimal points = $1 + 1 = 2$.

Place the decimal point two places from the right in the product 30125.

The result is $301.25$.

The cost of $2.5\text{ kg}$ of apples is $\textsf{₹ }301.25$.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

To divide a decimal number by a decimal number, we first convert the divisor into a whole number by multiplying both the dividend and the divisor by a power of 10.

The expression is $5.6 \div 0.07$.

The divisor is $0.07$. To make it a whole number, we need to multiply by 100 (since there are two decimal places).

Multiply both the dividend and the divisor by 100:

Dividend: $5.6 \times 100 = 560$

Divisor: $0.07 \times 100 = 7$

Now, the division becomes $560 \div 7$.

$560 \div 7 = 80$

So, $5.6 \div 0.07 = 80$.

Comparing this result with the given options, we find that option (B) matches our result.

The correct option is (B).

We will evaluate each calculation to determine which one is incorrect.

(A) $\frac{1}{4} \times \frac{1}{2}$

To multiply fractions, we multiply the numerators and the denominators:

$\frac{1}{4} \times \frac{1}{2} = \frac{1 \times 1}{4 \times 2} = \frac{1}{8}$

This calculation is correct.

(B) $\frac{3}{5} \div \frac{1}{5}$

To divide by a fraction, we multiply by its reciprocal. The reciprocal of $\frac{1}{5}$ is $\frac{5}{1}$ or 5.

$\frac{3}{5} \div \frac{1}{5} = \frac{3}{5} \times 5 = \frac{3 \times \cancel{5}}{\cancel{5}} = 3$

This calculation is correct.

(C) $0.1 + 0.01$

To add decimals, we align the decimal points and add column by column. We can write $0.1$ as $0.10$ to have the same number of decimal places as $0.01$.

$\begin{array}{cc} & 0 . 1 & 0 \\ + & 0 . 0 & 1 \\ \hline & 0 . 1 & 1 \\ \hline \end{array}$

$0.1 + 0.01 = 0.11$

This calculation is correct.

(D) $2.5 - 1.5$

To subtract decimals, we align the decimal points and subtract column by column.

$\begin{array}{cc} & 2 . 5 \\ - & 1 . 5 \\ \hline & 1 . 0 \\ \hline \end{array}$

$2.5 - 1.5 = 1.0 = 1$

The option states that $2.5 - 1.5 = 0.1$. This is incorrect.

The incorrect calculation is (D).

The correct option is (D).

We need to evaluate the expression $2\frac{1}{3} \times \frac{3}{7} + \frac{1}{2}$.

According to the order of operations (BODMAS/PEMDAS), we perform multiplication before addition.

First, convert the mixed fraction $2\frac{1}{3}$ to an improper fraction:

$2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$

Now, perform the multiplication $\frac{7}{3} \times \frac{3}{7}$:

$\frac{7}{3} \times \frac{3}{7} = \frac{7 \times 3}{3 \times 7} = \frac{21}{21}$

Simplify the product $\frac{21}{21}$:

$\frac{21}{21} = 1$

Now, perform the addition: $1 + \frac{1}{2}$.

$1 + \frac{1}{2} = 1\frac{1}{2}$

As an improper fraction, this is $1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}$.

The result of the calculation is $1\frac{1}{2}$.

Let's examine the options provided within the question:

• (A) $1 + \frac{1}{2} = 1\frac{1}{2}$: This shows the addition step and the correct final answer, assuming the multiplication result was 1.
• (B) $\frac{7}{3} \times \frac{3}{7} + \frac{1}{2} = 1 + \frac{1}{2} = 1\frac{1}{2}$: This shows the conversion of the mixed fraction, the multiplication resulting in 1, the addition step, and the correct final answer.
• (C) $\frac{21}{21} + \frac{1}{2} = 1 + \frac{1}{2} = 1\frac{1}{2}$: This shows the unsimplified result of the multiplication ($\frac{21}{21}$), its simplification to 1, the addition step, and the correct final answer.

All three options (A), (B), and (C) correctly show steps that lead to the correct final answer $1\frac{1}{2}$. Option (D) states that all of the above show the correct calculation and answer.

Therefore, the correct option is (D).

The average (mean) of a set of data is calculated by dividing the sum of all the values by the number of values.

Given data (heights in meters):

$1.5, 1.2, 1.8, 1.6, 1.9, 1.1, 1.7, 2.0, 1.4, 1.3,$

$1.7, 1.5, 1.9, 1.8, 1.3, 1.6, 1.4, 1.2, 2.1, 1.5$

Number of plants = 20

Sum of the heights:

Sum = $1.5 + 1.2 + 1.8 + 1.6 + 1.9 + 1.1 + 1.7 + 2.0 + 1.4 + 1.3 + 1.7 + 1.5 + 1.9 + 1.8 + 1.3 + 1.6 + 1.4 + 1.2 + 2.1 + 1.5$

Sum = $(1.5 \times 4) + (1.2 \times 2) + (1.8 \times 2) + (1.6 \times 2) + (1.9 \times 2) + (1.1 \times 1) + (1.7 \times 2) + (2.0 \times 1) + (1.4 \times 2) + (1.3 \times 2) + (2.1 \times 1)$

Sum = $6.0 + 2.4 + 3.6 + 3.2 + 3.8 + 1.1 + 3.4 + 2.0 + 2.8 + 2.6 + 2.1$

Sum = $31.5$ meters

Average height = $\frac{\text{Sum of heights}}{\text{Number of plants}}$

Average height = $\frac{31.5}{20}$

Performing the division:

$31.5 \div 20 = 1.575$

$\begin{array}{r} 1.575 \\ 20{\overline{\smash{\big)}\,31.500}} \\ \underline{-~\phantom{(}20\phantom{..00)}} \\ 11.50 \\ \underline{-~\phantom{(}10.00)} \\ 1.500 \\ \underline{-~\phantom{(}1.400)} \\ 0.100 \\ \underline{-~\phantom{(}0.100)} \\ 0.000 \end{array}$

The average height of the plants is $1.575\text{ m}$.

Now, we compare this result with the given options:

• (A) 1.55
• (B) 1.6
• (C) 1.65
• (D) 1.7

The calculated average $1.575$ is exactly halfway between $1.55$ and $1.60$. When rounding to one decimal place, $1.575$ rounds up to $1.6$. Among the given options, $1.6\text{ m}$ (option B) is the closest value to the calculated average.

The correct option is (B).

To convert a decimal to a fraction, we write the decimal as a fraction with the decimal part as the numerator and a power of 10 as the denominator. The power of 10 is determined by the number of decimal places.

The given decimal is $0.075$.

There are 3 digits after the decimal point (0, 7, and 5).

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

The numerator will be the decimal part without the leading zeros, which is 75.

The fraction is $\frac{75}{1000}$.

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

Let's find the GCD of 75 and 1000.

We can divide by common factors. Both are divisible by 5.

$\frac{75 \div 5}{1000 \div 5} = \frac{15}{200}$

Now, simplify $\frac{15}{200}$. Both are divisible by 5 again.

$\frac{15 \div 5}{200 \div 5} = \frac{3}{40}$

The numerator is 3 and the denominator is 40. The only common factor of 3 and 40 is 1. So, the fraction $\frac{3}{40}$ is in its simplest form.

Comparing this result with the given options, we find that option (C) matches our result.

Let's quickly check the other options:

• (A) $\frac{75}{1000}$: This is the fraction before simplification.
• (B) $\frac{15}{200}$: This is an intermediate simplified form, but not the simplest form.
• (D) $\frac{3}{80}$: This is not equivalent to $\frac{75}{1000}$ or $0.075$. ($\frac{3}{80} = 0.0375$)

The simplest form of $0.075$ as a fraction is $\frac{3}{40}$.

The correct option is (C).

To find the number of pieces that can be cut, we need to divide the total length of the wire by the length of each piece.

Total length of wire = $10\text{ m}$

Length of each piece = $0.5\text{ m}$

Number of pieces = Total length $\div$ Length of each piece

Number of pieces = $10 \div 0.5$

To divide by a decimal, we can convert the divisor into a whole number by multiplying both the dividend and the divisor by a power of 10. The divisor is $0.5$, which has one decimal place, so we multiply by 10.

Dividend: $10 \times 10 = 100$

Divisor: $0.5 \times 10 = 5$

The division becomes $100 \div 5$.

$100 \div 5 = 20$

So, 20 pieces of wire, each $0.5\text{ m}$ long, can be cut from a $10\text{ m}$ long wire.

Alternatively, we can write $0.5$ as a fraction: $0.5 = \frac{5}{10} = \frac{1}{2}$.

The division becomes $10 \div \frac{1}{2}$.

To divide by a fraction, multiply by its reciprocal. The reciprocal of $\frac{1}{2}$ is 2.

$10 \div \frac{1}{2} = 10 \times 2 = 20$.

The number of pieces is 20.

Comparing this result with the given options, we find that option (C) matches our result.

The correct option is (C).

Let's evaluate each statement based on mathematical definitions.

Statement 1: The product of a proper fraction and an improper fraction is always greater than the proper fraction.

A proper fraction $\frac{a}{b}$ has $0 < a < b$. An improper fraction $\frac{c}{d}$ has $c \ge d > 0$.

Consider a proper fraction, e.g., $\frac{1}{2}$. Consider an improper fraction that is equal to 1, e.g., $\frac{2}{2}$.

Their product is $\frac{1}{2} \times \frac{2}{2} = \frac{1}{2} \times 1 = \frac{1}{2}$.

The product ($\frac{1}{2}$) is not greater than the proper fraction ($\frac{1}{2}$). It is equal.

Since there is a case where the product is not strictly greater, Statement 1 is false.

Statement 2: Dividing a whole number by a unit fraction (like $\frac{1}{n}$) results in a value greater than the whole number.

A whole number is any of $0, 1, 2, 3, \dots$. A unit fraction $\frac{1}{n}$ has $n$ as a positive integer ($n \ge 1$).

Consider the whole number 5 and the unit fraction $\frac{1}{1} = 1$.

Dividing the whole number by the unit fraction: $5 \div 1 = 5$.

The result (5) is not greater than the whole number (5). It is equal.

Consider the whole number 0 and the unit fraction $\frac{1}{2}$.

Dividing the whole number by the unit fraction: $0 \div \frac{1}{2} = 0 \times 2 = 0$.

The result (0) is not greater than the whole number (0).

Since there are cases where the result is not strictly greater, Statement 2 is false.

Both Statement 1 and Statement 2 are false.

The correct option is (D).

The relationship between paise and rupees is that 1 Rupee is equal to 100 paise.

To convert an amount from paise to rupees, we divide the amount in paise by 100.

Given amount is 125 paise.

Amount in rupees = $\frac{\text{Amount in paise}}{100}$

Amount in rupees = $\frac{125}{100}$

Dividing 125 by 100 is equivalent to moving the decimal point in 125 two places to the left.

$125 \div 100 = 1.25$

So, 125 paise is equal to $\textsf{₹ }1.25$.

The blank should be filled with 1.25.

Comparing this result with the given options, we find that option (A) matches our result.

The correct option is (A).

To simplify the expression $10.5 - 2.3 \times 4 + 1.2$, we must follow the order of operations (BODMAS/PEMDAS): Multiplication and Division before Addition and Subtraction.

First, perform the multiplication:

$2.3 \times 4$

Multiply $23 \times 4 = 92$. Since $2.3$ has one decimal place, the result will have one decimal place.

$2.3 \times 4 = 9.2$

Now substitute this result back into the expression:

$10.5 - 9.2 + 1.2$

Now, perform the subtraction and addition from left to right.

First, subtract $9.2$ from $10.5$:

$10.5 - 9.2$

$\begin{array}{cc} & 10 . 5 \\ - & \phantom{0}9 . 2 \\ \hline & \phantom{0}1 . 3 \\ \hline \end{array}$

$10.5 - 9.2 = 1.3$

Now, add $1.2$ to the result:

$1.3 + 1.2$

$\begin{array}{cc} & 1 . 3 \\ + & 1 . 2 \\ \hline & 2 . 5 \\ \hline \end{array}$

$1.3 + 1.2 = 2.5$

The value of the expression is $2.5$.

Let's examine the options provided within the question:

• (A) $10.5 - 9.2 + 1.2 = 1.3 + 1.2 = 2.5$: This correctly follows the order of operations (multiplication first, then subtraction and addition from left to right) and arrives at the correct answer.
• (B) $10.5 - 9.2 + 1.2 = 10.5 - 10.4 = 0.1$: This performs the multiplication correctly ($2.3 \times 4 = 9.2$) but then adds $9.2 + 1.2$ before subtracting from $10.5$. This violates the left-to-right rule for addition and subtraction when they appear at the same level in the order of operations. $9.2 + 1.2 = 10.4$. $10.5 - 10.4 = 0.1$.
• (C) $10.5 - 2.3 \times 5.2 = 10.5 - 11.96 = -1.46$: This incorrectly adds $4 + 1.2$ first before multiplying.
• (D) $8.2 \times 4 + 1.2 = 32.8 + 1.2 = 34$: This incorrectly subtracts $10.5 - 2.3$ first before multiplying.

Option (A) shows the correct calculation steps and the correct answer.

The correct option is (A).

Let's classify the given fractions:

(a) $\frac{7}{9}$

Here, the numerator (7) is less than the denominator (9).

Therefore, $\frac{7}{9}$ is a proper fraction.

(b) $\frac{11}{5}$

Here, the numerator (11) is greater than the denominator (5).

Therefore, $\frac{11}{5}$ is an improper fraction.

(c) $3\frac{1}{4}$

This fraction consists of a whole number part (3) and a fractional part ($\frac{1}{4}$).

Therefore, $3\frac{1}{4}$ is a mixed fraction.

(d) $\frac{2}{2}$

Here, the numerator (2) is equal to the denominator (2).

Fractions where the numerator is greater than or equal to the denominator are classified as improper fractions.

Therefore, $\frac{2}{2}$ is an improper fraction.

To convert an improper fraction into a mixed fraction, we divide the numerator by the denominator.

The mixed fraction is written as Quotient $\frac{\text{Remainder}}{\text{Divisor}}$.

(a) $\frac{17}{4}$

Divide 17 by 4:

$17 \div 4 = 4$ with a remainder of 1.

Quotient = 4

Remainder = 1

Divisor = 4

So, the mixed fraction is $4\frac{1}{4}$.

(b) $\frac{29}{6}$

Divide 29 by 6:

$29 \div 6 = 4$ with a remainder of 5.

Quotient = 4

Remainder = 5

Divisor = 6

So, the mixed fraction is $4\frac{5}{6}$.

To convert a mixed fraction into an improper fraction, we use the formula:

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

(a) $4\frac{2}{3}$

Here, the whole number is 4, the numerator is 2, and the denominator is 3.

Improper fraction = $\frac{(4 \times 3) + 2}{3}$

= $\frac{12 + 2}{3}$

= $\frac{14}{3}$

So, $4\frac{2}{3}$ is equal to $\frac{14}{3}$.

(b) $1\frac{5}{7}$

Here, the whole number is 1, the numerator is 5, and the denominator is 7.

Improper fraction = $\frac{(1 \times 7) + 5}{7}$

= $\frac{7 + 5}{7}$

= $\frac{12}{7}$

So, $1\frac{5}{7}$ is equal to $\frac{12}{7}$.

(a) $\frac{3}{10} + \frac{4}{10} + \frac{1}{10}$

These are like fractions (same denominator). To add them, we add the numerators and keep the denominator the same.

Sum = $\frac{3 + 4 + 1}{10}$

= $\frac{8}{10}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

= $\frac{\cancel{8}^{4}}{\cancel{10}_{5}}$

= $\frac{4}{5}$

So, $\frac{3}{10} + \frac{4}{10} + \frac{1}{10} = \frac{4}{5}$.

(b) $\frac{2}{5} + \frac{1}{3}$

These are unlike fractions (different denominators). To add them, we first find a common denominator, which is the Least Common Multiple (LCM) of 5 and 3.

LCM(5, 3) = 15

Now, convert each fraction to an equivalent fraction with denominator 15.

For $\frac{2}{5}$: Multiply numerator and denominator by $\frac{15}{5} = 3$.

$\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}$

For $\frac{1}{3}$: Multiply numerator and denominator by $\frac{15}{3} = 5$.

$\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$

Now, add the equivalent like fractions:

$\frac{6}{15} + \frac{5}{15} = \frac{6 + 5}{15}$

= $\frac{11}{15}$

The fraction $\frac{11}{15}$ cannot be simplified further as 11 and 15 have no common factors other than 1.

So, $\frac{2}{5} + \frac{1}{3} = \frac{11}{15}$.

(a) $\frac{7}{8} - \frac{3}{8}$

These are like fractions (same denominator). To subtract them, we subtract the numerators and keep the denominator the same.

Difference = $\frac{7 - 3}{8}$

= $\frac{4}{8}$

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

= $\frac{\cancel{4}^{1}}{\cancel{8}_{2}}$

= $\frac{1}{2}$

So, $\frac{7}{8} - \frac{3}{8} = \frac{1}{2}$.

(b) $\frac{5}{6} - \frac{1}{4}$

These are unlike fractions (different denominators). To subtract them, we first find a common denominator, which is the Least Common Multiple (LCM) of 6 and 4.

Let's find the LCM of 6 and 4:

$\begin{array}{c|cc} 2 & 6 \;, & 4 \\ \hline 2 & 3 \; , & 2 \\ \hline 3 & 3 \; , & 1 \\ \hline & 1 \; , & 1 \end{array}$

LCM(6, 4) = $2 \times 2 \times 3 = 12$.

Now, convert each fraction to an equivalent fraction with denominator 12.

For $\frac{5}{6}$: Multiply numerator and denominator by $\frac{12}{6} = 2$.

$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

For $\frac{1}{4}$: Multiply numerator and denominator by $\frac{12}{4} = 3$.

$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

Now, subtract the equivalent like fractions:

$\frac{10}{12} - \frac{3}{12} = \frac{10 - 3}{12}$

= $\frac{7}{12}$

The fraction $\frac{7}{12}$ cannot be simplified further as 7 and 12 have no common factors other than 1.

So, $\frac{5}{6} - \frac{1}{4} = \frac{7}{12}$.

To simplify the expression $\frac{3}{4} + \frac{5}{6} - \frac{1}{2}$, we need to find a common denominator for the fractions. The denominators are 4, 6, and 2.

We find the Least Common Multiple (LCM) of 4, 6, and 2.

$\begin{array}{c|cc} 2 & 4 \;, & 6 \;, & 2 \\ \hline 2 & 2 \; , & 3 \; , & 1 \\ \hline 3 & 1 \; , & 3 \; , & 1 \\ \hline & 1 \; , & 1 \; , & 1 \end{array}$

The LCM of 4, 6, and 2 is $2 \times 2 \times 3 = 12$.

Now, we convert each fraction to an equivalent fraction with a denominator of 12.

For $\frac{3}{4}$: Multiply the numerator and denominator by $\frac{12}{4} = 3$.

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

For $\frac{5}{6}$: Multiply the numerator and denominator by $\frac{12}{6} = 2$.

$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

For $\frac{1}{2}$: Multiply the numerator and denominator by $\frac{12}{2} = 6$.

$\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}$

Now, we can perform the addition and subtraction with the equivalent fractions:

$\frac{9}{12} + \frac{10}{12} - \frac{6}{12} = \frac{9 + 10 - 6}{12}$

= $\frac{19 - 6}{12}$

= $\frac{13}{12}$

The fraction $\frac{13}{12}$ is an improper fraction. We can leave it as is, or convert it to a mixed fraction.

To convert to a mixed fraction, divide 13 by 12:

$13 \div 12 = 1$ with a remainder of 1.

So, $1\frac{1}{12}$.

The simplified form is $\frac{13}{12}$ or $1\frac{1}{12}$.

Thus, $\frac{3}{4} + \frac{5}{6} - \frac{1}{2} = \frac{13}{12}$.

(a) $7 \times \frac{3}{5}$

To multiply a whole number by a fraction, we can write the whole number as a fraction with denominator 1.

$7 \times \frac{3}{5} = \frac{7}{1} \times \frac{3}{5}$

Multiply the numerators together and the denominators together.

= $\frac{7 \times 3}{1 \times 5}$

= $\frac{21}{5}$

The product is $\frac{21}{5}$. This is an improper fraction.

(b) $\frac{4}{9} \times \frac{3}{8}$

To multiply fractions, we multiply the numerators and multiply the denominators.

$\frac{4}{9} \times \frac{3}{8} = \frac{4 \times 3}{9 \times 8}$

We can simplify by cancelling common factors before multiplying.

The numerator 4 and the denominator 8 have a common factor of 4.

The numerator 3 and the denominator 9 have a common factor of 3.

= $\frac{\cancel{4}^{1}}{\cancel{9}_{3}} \times \frac{\cancel{3}^{1}}{\cancel{8}_{2}}$

Now multiply the remaining numerators and denominators.

= $\frac{1 \times 1}{3 \times 2}$

= $\frac{1}{6}$

The product is $\frac{1}{6}$.

The phrase "$\frac{3}{5}$ of $10$ kg" means we need to calculate the product of $\frac{3}{5}$ and $10$ kg.

We can write this as:

Value = $\frac{3}{5} \times 10$ kg

To multiply a fraction by a whole number, we multiply the numerator by the whole number and keep the denominator.

Value = $\frac{3 \times 10}{5}$ kg

Value = $\frac{30}{5}$ kg

Now, we simplify the fraction $\frac{30}{5}$.

Divide 30 by 5:

$30 \div 5 = 6$

So, $\frac{30}{5} = 6$.

Therefore, $\frac{3}{5}$ of $10$ kg is $6$ kg.

The value of $\frac{3}{5}$ of $10$ kg is $6$ kg.

The reciprocal of a fraction is obtained by interchanging its numerator and denominator. For a non-zero number $a$, its reciprocal is $\frac{1}{a}$. For a fraction $\frac{a}{b}$, its reciprocal is $\frac{b}{a}$.

(a) $\frac{5}{11}$

To find the reciprocal of $\frac{5}{11}$, we interchange the numerator (5) and the denominator (11).

Reciprocal of $\frac{5}{11}$ is $\frac{11}{5}$.

The reciprocal of $\frac{5}{11}$ is $\frac{11}{5}$.

(b) $8$

We can write the whole number 8 as a fraction $\frac{8}{1}$.

To find the reciprocal of $\frac{8}{1}$, we interchange the numerator (8) and the denominator (1).

Reciprocal of $\frac{8}{1}$ is $\frac{1}{8}$.

The reciprocal of 8 is $\frac{1}{8}$.

(c) $1\frac{3}{4}$

First, convert the mixed fraction $1\frac{3}{4}$ into an improper fraction.

$1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}$.

Now, find the reciprocal of the improper fraction $\frac{7}{4}$ by interchanging the numerator (7) and the denominator (4).

Reciprocal of $\frac{7}{4}$ is $\frac{4}{7}$.

The reciprocal of $1\frac{3}{4}$ is $\frac{4}{7}$.

To divide a fraction by a whole number or another fraction, we multiply the first fraction by the reciprocal of the second number.

(a) $\frac{2}{3} \div 5$

We can write the whole number 5 as a fraction $\frac{5}{1}$.

So, the expression becomes $\frac{2}{3} \div \frac{5}{1}$.

The reciprocal of $\frac{5}{1}$ is $\frac{1}{5}$.

Now, we multiply $\frac{2}{3}$ by the reciprocal of $\frac{5}{1}$:

$\frac{2}{3} \div 5 = \frac{2}{3} \times \frac{1}{5}$

Multiply the numerators and the denominators:

= $\frac{2 \times 1}{3 \times 5}$

= $\frac{2}{15}$

The result is $\frac{2}{15}$.

(b) $\frac{5}{8} \div \frac{2}{3}$

The reciprocal of the second fraction $\frac{2}{3}$ is $\frac{3}{2}$.

Now, we multiply the first fraction $\frac{5}{8}$ by the reciprocal of the second fraction $\frac{3}{2}$:

$\frac{5}{8} \div \frac{2}{3} = \frac{5}{8} \times \frac{3}{2}$

Multiply the numerators and the denominators:

= $\frac{5 \times 3}{8 \times 2}$

= $\frac{15}{16}$

The result is $\frac{15}{16}$.

To write a number given in words as a decimal number, we identify the whole number part, the decimal point (indicated by 'and'), and the fractional part.

(a) Twenty-five and four-tenths

The whole number part is Twenty-five, which is 25.

The word 'and' indicates the decimal point.

The fractional part is four-tenths, which means $\frac{4}{10}$.

As a decimal, $\frac{4}{10} = 0.4$.

Combining the whole number part and the decimal part, we get $25 + 0.4 = 25.4$.

So, Twenty-five and four-tenths is written as 25.4.

(b) One hundred eight and ninety-two hundredths

The whole number part is One hundred eight, which is 108.

The word 'and' indicates the decimal point.

The fractional part is ninety-two hundredths, which means $\frac{92}{100}$.

As a decimal, $\frac{92}{100} = 0.92$.

Combining the whole number part and the decimal part, we get $108 + 0.92 = 108.92$.

So, One hundred eight and ninety-two hundredths is written as 108.92.

To convert a decimal number to a fraction, we write the decimal number as a fraction with the decimal part as the numerator and a power of 10 in the denominator (corresponding to the number of decimal places). Then we simplify the fraction to its simplest form.

(a) $0.6$

The decimal $0.6$ has one digit after the decimal point, which is in the tenths place. So, we can write it as 6 tenths.

$0.6 = \frac{6}{10}$

Now, simplify the fraction $\frac{6}{10}$. Both the numerator (6) and the denominator (10) are divisible by 2.

$\frac{6 \div 2}{10 \div 2} = \frac{3}{5}$

The simplest form of the fraction is $\frac{3}{5}$.

So, $0.6 = \frac{3}{5}$.

(b) $2.25$

The decimal $2.25$ has two digits after the decimal point, which are in the hundredths place. So, we can write it as 2 and 25 hundredths.

This can be written as a mixed number $2\frac{25}{100}$.

To convert this to an improper fraction, multiply the whole number (2) by the denominator (100) and add the numerator (25), keeping the denominator (100):

$2\frac{25}{100} = \frac{(2 \times 100) + 25}{100} = \frac{200 + 25}{100} = \frac{225}{100}$

Alternatively, $2.25$ can be read as 225 hundredths, which is $\frac{225}{100}$.

Now, simplify the fraction $\frac{225}{100}$. Both 225 and 100 are divisible by 25.

$225 \div 25 = 9$

$100 \div 25 = 4$

So, $\frac{225}{100} = \frac{9}{4}$.

The simplest form of the fraction is $\frac{9}{4}$.

So, $2.25 = \frac{9}{4}$. (This can also be written as the mixed number $2\frac{1}{4}$).

The place value of a digit in a decimal number depends on its position relative to the decimal point.

(a) In the decimal number $34.567$, the digit 6 is in the second place after the decimal point.

The first place after the decimal point is the tenths place ($\frac{1}{10}$).

The second place after the decimal point is the hundredths place ($\frac{1}{100}$).

So, the place value of 6 in $34.567$ is 6 hundredths, or $\frac{6}{100}$.

The place value of 6 is hundredths.

(b) In the decimal number $106.21$, the digit 6 is in the first place before the decimal point.

The first place before the decimal point is the ones place (or units place) ($10^0$).

So, the place value of 6 in $106.21$ is 6 ones, or $6 \times 1 = 6$.

The place value of 6 is ones.

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

(a) $\frac{3}{5}$

Method 1: By division

We divide $3$ by $5$:

$\begin{array}{r} 0.6\phantom{)} \\ 5{\overline{\smash{\big)}\,3.0\phantom{)}}} \\ \underline{-~\phantom{(}(30)} \\ 0\phantom{)} \end{array}$

So, $\frac{3}{5} = 0.6$.

Method 2: By converting denominator to a power of 10

We can convert the denominator $5$ to $10$ by multiplying both the numerator and denominator by $2$.

$\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}$

$\frac{6}{10} = 0.6$

Thus, $\frac{3}{5} = \mathbf{0.6}$.

(b) $\frac{9}{20}$

Method 1: By division

We divide $9$ by $20$:

$\begin{array}{r} 0.45\phantom{)} \\ 20{\overline{\smash{\big)}\,9.00\phantom{)}}} \\ \underline{-~\phantom{(}(80)}\phantom{0} \\ 100\phantom{)} \\ \underline{-~\phantom{()}(100)} \\ 0\phantom{)} \end{array}$

So, $\frac{9}{20} = 0.45$.

Method 2: By converting denominator to a power of 10

We can convert the denominator $20$ to $100$ by multiplying both the numerator and denominator by $5$.

$\frac{9}{20} = \frac{9 \times 5}{20 \times 5} = \frac{45}{100}$

$\frac{45}{100} = 0.45$

Thus, $\frac{9}{20} = \mathbf{0.45}$.

To compare decimals, we first compare the whole number parts. If they are equal, we compare the digits to the right of the decimal point, starting from the tenths place, then the hundredths place, and so on, until the digits are different.

(a) $0.5 \_\_\_ 0.05$

The whole number parts are both $0$.

Compare the tenths digits: $5$ in $0.5$ and $0$ in $0.05$.

Since $5 > 0$, we have $0.5 > 0.05$.

So, $0.5 \mathbf{>} 0.05$.

(b) $3.42 \_\_\_ 3.24$

The whole number parts are both $3$.

Compare the tenths digits: $4$ in $3.42$ and $2$ in $3.24$.

Since $4 > 2$, we have $3.42 > 3.24$.

So, $3.42 \mathbf{>} 3.24$.

(c) $1.7 \_\_\_ 1.70$

The whole number parts are both $1$.

Compare the tenths digits: $7$ in $1.7$ and $7$ in $1.70$. They are equal.

Compare the hundredths digits: $1.7$ can be written as $1.70$. The hundredths digit is $0$ in both numbers.

Since all corresponding digits are equal, the numbers are equal.

So, $1.7 \mathbf{=} 1.70$.

To find the product of decimals, we multiply the numbers ignoring the decimal points, and then place the decimal point in the product such that the number of decimal places in the product is the sum of the number of decimal places in the numbers being multiplied.

(a) $0.3 \times 0.8$

Multiply $3$ and $8$ ignoring the decimal points: $3 \times 8 = 24$.

$0.3$ has one decimal place.

$0.8$ has one decimal place.

The total number of decimal places in the product should be $1 + 1 = 2$.

So, place the decimal point two places from the right in $24$.

$0.3 \times 0.8 = \mathbf{0.24}$.

(b) $2.1 \times 1.5$

Multiply $21$ and $15$ ignoring the decimal points.

$\begin{array}{cc}& & 2 & 1 \\ \times & & 1 & 5 \\ \hline && 10 & 5 \\ & 2 & 1 & \times \\ \hline & 3 & 1 & 5 \\ \hline \end{array}$

$2.1$ has one decimal place.

$1.5$ has one decimal place.

The total number of decimal places in the product should be $1 + 1 = 2$.

So, place the decimal point two places from the right in $315$.

$2.1 \times 1.5 = \mathbf{3.15}$.

To find the quotient of decimals, we perform division. If the divisor is a decimal, we first convert it to a whole number by multiplying both the divisor and the dividend by a suitable power of 10.

(a) $14.4 \div 12$

We divide $14.4$ by $12$. The divisor is already a whole number.

Perform long division:

$\begin{array}{r} 1.2\phantom{)} \\ 12{\overline{\smash{\big)}\,14.4\phantom{)}}} \\ \underline{-~\phantom{(}(12)}\phantom{4} \\ 24\phantom{)} \\ \underline{-~\phantom{()}(24)} \\ 0\phantom{)} \end{array}$

So, $14.4 \div 12 = \mathbf{1.2}$.

(b) $3.9 \div 0.3$

We need to divide $3.9$ by $0.3$. The divisor $0.3$ is a decimal.

To make the divisor a whole number, we multiply both the dividend and the divisor by $10$.

$3.9 \times 10 = 39$

$0.3 \times 10 = 3$

Now we need to divide $39$ by $3$.

Perform long division:

$\begin{array}{r} 13\phantom{)} \\ 3{\overline{\smash{\big)}\,39\phantom{)}}} \\ \underline{-~\phantom{(}(3)}\phantom{9} \\ 09\phantom{)} \\ \underline{-~\phantom{()}(9)} \\ 0\phantom{)} \end{array}$

So, $3.9 \div 0.3 = \mathbf{13}$.

Given:

Distance covered by the car in $1$ liter of petrol $= 55.3$ km.

To Find:

Distance covered by the car in $10$ liters of petrol.

Solution:

To find the distance covered in $10$ liters of petrol, we need to multiply the distance covered in $1$ liter by $10$.

Distance covered in $10$ liters $= \text{Distance in } 1 \text{ liter } \times 10$

Distance covered in $10$ liters $= 55.3 \text{ km } \times 10$

Multiplying a decimal by $10$ involves shifting the decimal point one place to the right.

$55.3 \times 10 = 553.0$

So, the distance covered in $10$ liters is $553$ km.

The car will cover a distance of $\mathbf{553}$ km in $10$ liters of petrol.

Given:

Side of the regular polygon $= 3.5$ cm.

Perimeter of the regular polygon $= 17.5$ cm.

To Find:

The number of sides of the regular polygon.

Solution:

The perimeter of a regular polygon is the sum of the lengths of all its sides.

Since all sides of a regular polygon are equal in length, the perimeter is given by the formula:

Perimeter = Number of sides $\times$ Length of one side

Let $n$ be the number of sides of the polygon.

We have the equation:

$17.5 \text{ cm} = n \times 3.5 \text{ cm}$

To find the number of sides ($n$), we need to divide the perimeter by the length of one side.

$n = \frac{\text{Perimeter}}{\text{Length of one side}}$

$n = \frac{17.5}{3.5}$

To perform this division, we can remove the decimal points by multiplying both the numerator and the denominator by $10$.

$n = \frac{17.5 \times 10}{3.5 \times 10} = \frac{175}{35}$

Now, we perform the division of $175$ by $35$.

$\begin{array}{r} 5\phantom{)} \\ 35{\overline{\smash{\big)}\,175\phantom{)}}} \\ \underline{-~\phantom{(}(175)} \\ 0\phantom{)} \end{array}$

So, $n = 5$.

The polygon has $\mathbf{5}$ sides.

A regular polygon with $5$ sides is called a regular pentagon.

To simplify the expression, we need to follow the order of operations (PEMDAS/BODMAS), which means we perform the operations inside the parentheses first, then any multiplication or division from left to right, and finally addition or subtraction from left to right.

First, evaluate the multiplication part: $(0.5 \times 10)$.

$0.5 \times 10 = 5$

Multiplying a decimal by $10$ shifts the decimal point one place to the right.

Next, evaluate the division part: $(2.4 \div 0.4)$.

To divide by a decimal, we convert the divisor into a whole number by multiplying both the dividend and the divisor by a suitable power of $10$. In this case, multiply by $10$.

$2.4 \div 0.4 = \frac{2.4}{0.4} = \frac{2.4 \times 10}{0.4 \times 10} = \frac{24}{4}$

Now perform the division:

$\begin{array}{r} 6\phantom{)} \\ 4{\overline{\smash{\big)}\,24\phantom{)}}} \\ \underline{-~\phantom{(}(24)} \\ 0\phantom{)} \end{array}$

So, $2.4 \div 0.4 = 6$.

Finally, add the results from the multiplication and division parts.

Expression $= (0.5 \times 10) + (2.4 \div 0.4)$

Expression $= 5 + 6$

$5 + 6 = 11$

The simplified value of the expression is $\mathbf{11}$.

Given:

Total number of students in the class $= 40$.

Fraction of boys in the class $= \frac{3}{5}$.

To Find:

The number of girls in the class.

Solution:

First, we find the number of boys in the class.

Number of boys = Fraction of boys $\times$ Total number of students

Number of boys $= \frac{3}{5} \times 40$

Number of boys $= \frac{3 \times 40}{5}$

We can simplify the fraction by dividing $40$ by $5$:

Number of boys $= 3 \times \frac{\cancel{40}^{8}}{\cancel{5}_{1}}$

Number of boys $= 3 \times 8$

Number of boys $= 24$

So, there are $24$ boys in the class.

Now, we can find the number of girls by subtracting the number of boys from the total number of students.

Number of girls = Total number of students - Number of boys

Number of girls $= 40 - 24$

Number of girls $= 16$

Thus, there are $\mathbf{16}$ girls in the class.

To find the product of mixed fractions, we first convert the mixed fractions into improper fractions.

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

$2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$

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, multiply the improper fractions:

$2\frac{1}{3} \times 1\frac{1}{4} = \frac{7}{3} \times \frac{5}{4}$

To multiply fractions, we multiply the numerators together and the denominators together:

$\frac{7}{3} \times \frac{5}{4} = \frac{7 \times 5}{3 \times 4} = \frac{35}{12}$

The product is $\frac{35}{12}$.

We can also express the answer as a mixed fraction by dividing the numerator by the denominator.

$35 \div 12$. $12 \times 2 = 24$, $12 \times 3 = 36$. The quotient is $2$ and the remainder is $35 - 24 = 11$.

So, $\frac{35}{12} = 2\frac{11}{12}$.

The value of the expression is $\mathbf{\frac{35}{12}}$ or $\mathbf{2\frac{11}{12}}$.

Given:

Distance covered by the car in $1$ liter of petrol $= 16$ km.

Quantity of petrol used $= 2\frac{3}{4}$ liters.

To Find:

The total distance covered by the car using $2\frac{3}{4}$ liters of petrol.

Solution:

The distance covered is directly proportional to the amount of petrol used.

Total distance covered = Distance covered in $1$ liter $\times$ Total liters of petrol

Total distance covered $= 16 \text{ km} \times 2\frac{3}{4}$

First, convert the mixed fraction $2\frac{3}{4}$ into an improper fraction.

$2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$

Now, multiply the distance per liter by the improper fraction representing the total liters.

Total distance covered $= 16 \times \frac{11}{4}$

We can write $16$ as $\frac{16}{1}$.

Total distance covered $= \frac{16}{1} \times \frac{11}{4}$

Multiply the numerators and the denominators.

Total distance covered $= \frac{16 \times 11}{1 \times 4}$

Before multiplying, we can simplify by cancelling common factors between the numerator and the denominator.

We can cancel $16$ in the numerator and $4$ in the denominator, as $16$ is divisible by $4$.

$\frac{\cancel{16}^{4} \times 11}{1 \times \cancel{4}_{1}}$

Now, multiply the remaining numbers.

Total distance covered $= \frac{4 \times 11}{1 \times 1} = \frac{44}{1} = 44$

The total distance covered by the car is $44$ km.

The car will cover $\mathbf{44}$ km using $2\frac{3}{4}$ liters of petrol.

Given:

Perimeter of the rectangle $= 13$ cm.

Width of the rectangle ($w$) $= 2\frac{3}{4}$ cm.

To Find:

The length of the rectangle ($l$).

Solution:

The formula for the perimeter of a rectangle is given by:

$P = 2(l + w)$

Substitute the given values into the formula:

$13 = 2\left(l + 2\frac{3}{4}\right)$

First, convert the mixed fraction width into an improper fraction:

$2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$

Substitute this improper fraction back into the perimeter equation:

$13 = 2\left(l + \frac{11}{4}\right)$

Divide both sides of the equation by $2$:

$\frac{13}{2} = l + \frac{11}{4}$

To find the length ($l$), subtract the width ($\frac{11}{4}$) from $\frac{13}{2}$:

$l = \frac{13}{2} - \frac{11}{4}$

To subtract these fractions, find a common denominator, which is $4$. Convert $\frac{13}{2}$ to an equivalent fraction with a denominator of $4$:

$\frac{13}{2} = \frac{13 \times 2}{2 \times 2} = \frac{26}{4}$

Now perform the subtraction:

$l = \frac{26}{4} - \frac{11}{4}$

$l = \frac{26 - 11}{4}$

$l = \frac{15}{4}$

The length of the rectangle is $\frac{15}{4}$ cm.

We can also express the length as a mixed number:

$\frac{15}{4} = 3\frac{3}{4}$

The length of the rectangle is $\mathbf{\frac{15}{4}}$ cm or $\mathbf{3\frac{3}{4}}$ cm.

To Find:

The amount by which $32.6$ should be decreased to obtain $15.8$.

Solution:

Let the unknown amount be $x$.

According to the question, if $32.6$ is decreased by $x$, the result is $15.8$. This can be written as an equation:

$32.6 - x = 15.8$

To find the value of $x$, we need to rearrange the equation. We can add $x$ to both sides and subtract $15.8$ from both sides:

$32.6 - 15.8 = x$

So, $x = 32.6 - 15.8$.

Now, we perform the subtraction:

$\begin{array}{cc} & 3 & 2 & . & 6 \\ - & 1 & 5 & . & 8 \\ \hline & 1 & 6 & . & 8 \\ \hline \end{array}$

$32.6 - 15.8 = 16.8$

Thus, $x = 16.8$.

$32.6$ should be decreased by $\mathbf{16.8}$ to get $15.8$.

Solution:

To simplify the expression, we first convert the mixed fractions into improper fractions.

Convert $3\frac{1}{5}$ to an improper fraction:

$3\frac{1}{5} = \frac{(3 \times 5) + 1}{5} = \frac{15 + 1}{5} = \frac{16}{5}$

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

$2\frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}$

Convert $1\frac{1}{4}$ to an improper fraction:

$1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4}$

The expression becomes:

$\frac{16}{5} + \frac{8}{3} - \frac{5}{4}$

Next, we find the Least Common Multiple (LCM) of the denominators $5, 3,$ and $4$ to add and subtract the fractions.

The prime factorization of the denominators are:

$5 = 5$

$3 = 3$

$4 = 2 \times 2 = 2^2$

The LCM is the product of the highest powers of all prime factors involved.

LCM$(5, 3, 4) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$.

Now, we convert each fraction to an equivalent fraction with a denominator of $60$.

$\frac{16}{5} = \frac{16 \times 12}{5 \times 12} = \frac{192}{60}$

$\frac{8}{3} = \frac{8 \times 20}{3 \times 20} = \frac{160}{60}$

$\frac{5}{4} = \frac{5 \times 15}{4 \times 15} = \frac{75}{60}$

So the expression is:

$\frac{192}{60} + \frac{160}{60} - \frac{75}{60}$

Now, we perform the addition and subtraction on the numerators:

$\frac{192 + 160 - 75}{60}$

First, add $192$ and $160$:

$192 + 160 = 352$

Then, subtract $75$ from $352$:

$\begin{array}{cc} & 3 & 5 & 2 \\ - & & 7 & 5 \\ \hline & 2 & 7 & 7 \\ \hline \end{array}$

So, $192 + 160 - 75 = 277$.

The simplified fraction is $\frac{277}{60}$.

We can also express this improper fraction as a mixed number by dividing the numerator by the denominator.

$277 \div 60$

$277 = 4 \times 60 + 37$

So, the quotient is $4$ and the remainder is $37$.

$\frac{277}{60} = 4\frac{37}{60}$

The simplified value is $\mathbf{\frac{277}{60}}$ or $\mathbf{4\frac{37}{60}}$.

Given:

Weight of apples $= 2\frac{1}{2}$ kg.

Weight of grapes $= 1\frac{3}{4}$ kg.

Weight of mangoes $= \frac{3}{8}$ kg.

To Find:

The total weight of fruits Manoj bought.

Solution:

To find the total weight, we need to add the weights of apples, grapes, and mangoes.

Total weight = Weight of apples + Weight of grapes + Weight of mangoes

Total weight $= 2\frac{1}{2} + 1\frac{3}{4} + \frac{3}{8}$

First, convert the mixed fractions to improper fractions:

$2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$

$1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}$

So the expression becomes:

Total weight $= \frac{5}{2} + \frac{7}{4} + \frac{3}{8}$

To add these fractions, we find the LCM of the denominators $2, 4,$ and $8$.

The LCM of $2, 4,$ and $8$ is $8$.

Now, convert each fraction to an equivalent fraction with a denominator of $8$.

$\frac{5}{2} = \frac{5 \times 4}{2 \times 4} = \frac{20}{8}$

$\frac{7}{4} = \frac{7 \times 2}{4 \times 2} = \frac{14}{8}$

$\frac{3}{8}$ remains the same.

Now, add the equivalent fractions:

Total weight $= \frac{20}{8} + \frac{14}{8} + \frac{3}{8}$

Total weight $= \frac{20 + 14 + 3}{8}$

Total weight $= \frac{34 + 3}{8}$

Total weight $= \frac{37}{8}$

The total weight of fruits is $\frac{37}{8}$ kg.

We can express this as a mixed number by dividing $37$ by $8$.

$37 \div 8 = 4$ with a remainder of $5$.

So, $\frac{37}{8} = 4\frac{5}{8}$.

The total weight of fruits Manoj bought is $\mathbf{\frac{37}{8}}$ kg or $\mathbf{4\frac{5}{8}}$ kg.

Given:

Length of the rectangular sheet ($l$) $= 12\frac{1}{2}$ cm.

Width of the rectangular sheet ($w$) $= 10\frac{2}{3}$ cm.

To Find:

The perimeter and the area of the rectangular sheet.

Solution:

First, convert the mixed fractions to improper fractions.

Length $l = 12\frac{1}{2} = \frac{(12 \times 2) + 1}{2} = \frac{24 + 1}{2} = \frac{25}{2}$ cm.

Width $w = 10\frac{2}{3} = \frac{(10 \times 3) + 2}{3} = \frac{30 + 2}{3} = \frac{32}{3}$ cm.

Calculate the Perimeter of the rectangle.

The formula for the perimeter of a rectangle is $P = 2(l + w)$.

$P = 2\left(\frac{25}{2} + \frac{32}{3}\right)$

To add the fractions inside the parentheses, find the LCM of the denominators $2$ and $3$. The LCM is $6$.

$P = 2\left(\frac{25 \times 3}{2 \times 3} + \frac{32 \times 2}{3 \times 2}\right)$

$P = 2\left(\frac{75}{6} + \frac{64}{6}\right)$

$P = 2\left(\frac{75 + 64}{6}\right)$

$P = 2\left(\frac{139}{6}\right)$

$P = \frac{2 \times 139}{6}$

Cancel the common factor $2$ from the numerator and the denominator:

$P = \frac{\cancel{2}^{1} \times 139}{\cancel{6}_{3}}$

$P = \frac{139}{3}$ cm.

The perimeter is $\frac{139}{3}$ cm.

We can also write this as a mixed number: $139 \div 3$. $139 = 3 \times 46 + 1$.

$P = 46\frac{1}{3}$ cm.

The perimeter of the rectangular sheet is $\mathbf{\frac{139}{3}}$ cm or $\mathbf{46\frac{1}{3}}$ cm.

Calculate the Area of the rectangle.

The formula for the area of a rectangle is $A = l \times w$.

$A = \frac{25}{2} \times \frac{32}{3}$

$A = \frac{25 \times 32}{2 \times 3}$

Cancel the common factor $2$ from $32$ in the numerator and $2$ in the denominator:

$A = \frac{25 \times \cancel{32}^{16}}{\cancel{2}_{1} \times 3}$

$A = \frac{25 \times 16}{1 \times 3}$

$A = \frac{400}{3}$ cm$^2$.

The area is $\frac{400}{3}$ cm$^2$.

We can also write this as a mixed number: $400 \div 3$. $400 = 3 \times 133 + 1$.

$A = 133\frac{1}{3}$ cm$^2$.

The area of the rectangular sheet is $\mathbf{\frac{400}{3}}$ cm$^2$ or $\mathbf{133\frac{1}{3}}$ cm$^2$.

Given:

Time taken by Saili to finish the whole book $= 1\frac{1}{4}$ hours.

Fraction of the book to be finished $= \frac{3}{5}$.

To Find:

The time taken by Saili to finish $\frac{3}{5}$ part of the book.

Solution:

First, convert the time taken to finish the whole book from a mixed fraction to an improper fraction.

$1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4}$ hours.

The time taken to finish a part of the book is the product of the time taken to finish the whole book and the fraction of the book to be finished.

Time taken for $\frac{3}{5}$ part = (Time for whole book) $\times$ (Fraction of book)

Time taken for $\frac{3}{5}$ part $= \frac{5}{4} \times \frac{3}{5}$

To multiply the fractions, multiply the numerators together and the denominators together:

Time taken for $\frac{3}{5}$ part $= \frac{5 \times 3}{4 \times 5}$

We can cancel the common factor $5$ from the numerator and the denominator:

Time taken for $\frac{3}{5}$ part $= \frac{\cancel{5}^{1} \times 3}{4 \times \cancel{5}_{1}}$

Time taken for $\frac{3}{5}$ part $= \frac{1 \times 3}{4 \times 1} = \frac{3}{4}$ hours.

The time taken to finish $\frac{3}{5}$ part of the book is $\frac{3}{4}$ hours.

This can also be expressed in minutes. Since $1$ hour $= 60$ minutes, $\frac{3}{4}$ hours $= \frac{3}{4} \times 60$ minutes.

$\frac{3}{4} \times 60 = 3 \times \frac{60}{4} = 3 \times 15 = 45$ minutes.

Saili will take $\mathbf{\frac{3}{4}}$ hours or $\mathbf{45}$ minutes to finish $\frac{3}{5}$ part of the book.

Given:

Total length of the rope $= 13\frac{1}{2}$ meters.

Number of equal pieces the rope is cut into $= 9$.

To Find:

The length of each piece of the rope.

Solution:

To find the length of each piece, we need to divide the total length of the rope by the number of equal pieces.

Length of each piece = Total length $\div$ Number of pieces

Length of each piece $= 13\frac{1}{2} \div 9$

First, convert the mixed fraction $13\frac{1}{2}$ into an improper fraction.

$13\frac{1}{2} = \frac{(13 \times 2) + 1}{2} = \frac{26 + 1}{2} = \frac{27}{2}$

So the expression becomes:

Length of each piece $= \frac{27}{2} \div 9$

Dividing by a whole number is the same as multiplying by its reciprocal. The reciprocal of $9$ is $\frac{1}{9}$.

Length of each piece $= \frac{27}{2} \times \frac{1}{9}$

Multiply the numerators together and the denominators together:

Length of each piece $= \frac{27 \times 1}{2 \times 9}$

We can simplify by cancelling common factors. $27$ and $9$ have a common factor of $9$.

Length of each piece $= \frac{\cancel{27}^{3} \times 1}{2 \times \cancel{9}_{1}}$

Length of each piece $= \frac{3 \times 1}{2 \times 1} = \frac{3}{2}$ meters.

The length of each piece is $\frac{3}{2}$ meters.

We can also express this as a mixed number or a decimal.

As a mixed number: $\frac{3}{2} = 1\frac{1}{2}$ meters.

As a decimal: $\frac{3}{2} = 1.5$ meters.

The length of each piece is $\mathbf{\frac{3}{2}}$ meters or $\mathbf{1\frac{1}{2}}$ meters or $\mathbf{1.5}$ meters.

Given:

The numbers are $\frac{2}{5}, \frac{3}{4}, 0.8, 0.75, \frac{1}{2}$.

To Convert and Arrange:

Convert the given numbers into decimals and arrange them in ascending order.

Solution:

We need to convert the fractions into decimals by dividing the numerator by the denominator.

Convert $\frac{2}{5}$ to a decimal:

Divide $2$ by $5$.

$\begin{array}{r} 0.4\phantom{)} \\ 5{\overline{\smash{\big)}\,2.0\phantom{)}}} \\ \underline{-~\phantom{(}(20)} \\ 0\phantom{)} \end{array}$

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

Convert $\frac{3}{4}$ to a decimal:

Divide $3$ by $4$.

$\begin{array}{r} 0.75\phantom{)} \\ 4{\overline{\smash{\big)}\,3.00\phantom{)}}} \\ \underline{-~\phantom{(}(28)}\phantom{0} \\ 20\phantom{)} \\ \underline{-~\phantom{()}(20)} \\ 0\phantom{)} \end{array}$

So, $\frac{3}{4} = 0.75$.

Convert $\frac{1}{2}$ to a decimal:

Divide $1$ by $2$.

$\begin{array}{r} 0.5\phantom{)} \\ 2{\overline{\smash{\big)}\,1.0\phantom{)}}} \\ \underline{-~\phantom{(}(10)} \\ 0\phantom{)} \end{array}$

So, $\frac{1}{2} = 0.5$.

The given numbers in decimal form are:

$\frac{2}{5} = 0.4$

$\frac{3}{4} = 0.75$

$0.8 = 0.8$

$0.75 = 0.75$

$\frac{1}{2} = 0.5$

Now, we arrange these decimal values in ascending order:

$0.4, 0.5, 0.75, 0.75, 0.8$

Comparing the decimal values, we have:

$0.4 < 0.5 < 0.75 = 0.75 < 0.8$

Mapping these decimal values back to the original numbers, the ascending order is:

$\frac{2}{5}, \frac{1}{2}, \frac{3}{4}, 0.75, 0.8$

The numbers arranged in ascending order are: $\mathbf{\frac{2}{5}, \frac{1}{2}, \frac{3}{4}, 0.75, 0.8}$.

Given:

Cost of $1$ meter of cloth $= \textsf{₹}125.75$.

Length of cloth purchased $= 8.5$ meters.

To Find:

The total cost of $8.5$ meters of cloth.

Solution:

To find the total cost of the cloth, we need to multiply the cost of $1$ meter of cloth by the total length of cloth purchased.

Total cost = Cost per meter $\times$ Length of cloth

Total cost $= \textsf{₹}125.75 \times 8.5$

We multiply $125.75$ by $8.5$.

Multiply the numbers ignoring the decimal points: $12575 \times 85$.

$\begin{array}{ccccccc}& & 1 & 2 & 5 & 7 & 5 \\ \times & & & & & 8 & 5 \\ \hline && 6 & 2 & 8 & 7 & 5 \\ & 10 & 0 & 6 & 0 & 0 & \times \\ \hline 1 & 0 & 6 & 8 & 8 & 7 & 5 \\ \hline \end{array}$

The number $125.75$ has $2$ decimal places.

The number $8.5$ has $1$ decimal place.

The total number of decimal places in the product is $2 + 1 = 3$.

So, we place the decimal point $3$ places from the right in the product $1068875$.

Total cost $= 1068.875$

Since cost is usually expressed in two decimal places (for paise), $\textsf{₹}1068.875$ can be rounded to $\textsf{₹}1068.88$. However, if the question doesn't specify rounding, we can keep the exact value.

The cost of $8.5$ meters of cloth is $\textsf{₹}\mathbf{1068.875}$.

Given:

Total distance covered by the car $= 259.2$ km.

Time taken to cover the distance $= 4$ hours.

To Find:

The average distance covered by the car in $1$ hour.

Solution:

To find the average distance covered by the car in $1$ hour, we need to divide the total distance covered by the time taken.

Average distance per hour = $\frac{\text{Total Distance}}{\text{Time Taken}}$

Average distance per hour $= \frac{259.2 \text{ km}}{4 \text{ hours}}$

We need to calculate $259.2 \div 4$.

Perform the long division:

$\begin{array}{r} 64.8\phantom{)} \\ 4{\overline{\smash{\big)}\,259.2\phantom{)}}} \\ \underline{-~\phantom{(}(24)}\phantom{9.2} \\ 19\phantom{.2} \\ \underline{-~\phantom{()}(16)}\phantom{.2} \\ 32\phantom{)} \\ \underline{-~\phantom{()}(32)} \\ 0\phantom{)} \end{array}$

So, $259.2 \div 4 = 64.8$.

The average distance covered by the car in $1$ hour is $64.8$ km.

The average distance covered by the car in $1$ hour is $\mathbf{64.8}$ km.

To simplify the expression $(5.6 \times 2.5) + (15.3 \div 0.9) - 10.5$, we follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Given expression: $(5.6 \times 2.5) + (15.3 \div 0.9) - 10.5$

Step 1: Evaluate the operations inside the parentheses.

First, calculate the multiplication $(5.6 \times 2.5)$.

Multiply $56$ by $25$:

$\begin{array}{cc}& & 5 & 6 \\ \times & & 2 & 5 \\ \hline && 2 & 8 & 0 \\ & 1 & 1 & 2 & \times \\ \hline 1 & 4 & 0 & 0 \\ \hline \end{array}$

Since $5.6$ has $1$ decimal place and $2.5$ has $1$ decimal place, the product will have $1 + 1 = 2$ decimal places.

$5.6 \times 2.5 = 14.00 = 14$.

Next, calculate the division $(15.3 \div 0.9)$.

To divide $15.3$ by $0.9$, we make the divisor a whole number by multiplying both the dividend and the divisor by $10$.

$15.3 \div 0.9 = \frac{15.3 \times 10}{0.9 \times 10} = \frac{153}{9}$.

Now, divide $153$ by $9$:

$\begin{array}{r} 17\phantom{)} \\ 9{\overline{\smash{\big)}\,153\phantom{)}}} \\ \underline{-~\phantom{(}(9)}\phantom{3} \\ 63\phantom{)} \\ \underline{-~\phantom{()}(63)} \\ 0\phantom{)} \end{array}$

So, $15.3 \div 0.9 = 17$.

Step 2: Substitute the results back into the original expression.

The expression becomes: $14 + 17 - 10.5$.

Step 3: Perform addition and subtraction from left to right.

First, perform the addition: $14 + 17$.

$14 + 17 = 31$.

The expression becomes: $31 - 10.5$.

Step 4: Perform the subtraction: $31 - 10.5$.

$\begin{array}{cc} & 3 & 1 & . & 0 \\ - & 1 & 0 & . & 5 \\ \hline & 2 & 0 & . & 5 \\ \hline \end{array}$

$31 - 10.5 = 20.5$.

The simplified value of the expression is $\mathbf{20.5}$.

Given:

Initial amount of water in the bucket $= 20.5$ liters.

Amount of water used $= 4.75$ liters.

Amount of water added $= 5.8$ liters.

To Find:

The amount of water in the bucket now.

Solution:

We start with the initial amount of water, subtract the amount used, and then add the amount added.

Amount of water after using $4.75$ liters $= 20.5 - 4.75$

Perform the subtraction:

$\begin{array}{cc} & 2 & 0 & . & 5 & 0 \\ - & & 4 & . & 7 & 5 \\ \hline & 1 & 5 & . & 7 & 5 \\ \hline \end{array}$

So, $20.5 - 4.75 = 15.75$ liters.

Now, $5.8$ liters of water is added to the bucket.

Amount of water now $= 15.75 + 5.8$

Perform the addition:

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

So, $15.75 + 5.8 = 21.55$ liters.

The amount of water in the bucket now is $\mathbf{21.55}$ liters.

Given:

Part of the apple eaten by Ritu $= \frac{3}{5}$.

The remaining part was eaten by Somu.

To Find:

The part of the apple eaten by Somu.

Who ate more and by how much.

Solution:

Let the whole apple be represented by $1$.

The part of the apple eaten by Somu is the remaining part after Ritu ate her share.

Part eaten by Somu = Whole apple - Part eaten by Ritu

Part eaten by Somu $= 1 - \frac{3}{5}$

To subtract the fraction from $1$, we can write $1$ as $\frac{5}{5}$ (since the denominator of the fraction is $5$).

Part eaten by Somu $= \frac{5}{5} - \frac{3}{5}$

Now, subtract the numerators as the denominators are the same.

Part eaten by Somu $= \frac{5 - 3}{5} = \frac{2}{5}$

So, Somu ate $\mathbf{\frac{2}{5}}$ part of the apple.

Now, we need to compare the parts eaten by Ritu and Somu to see who ate more.

Ritu ate $\frac{3}{5}$ part.

Somu ate $\frac{2}{5}$ part.

Since the denominators are the same ($5$), we compare the numerators.

Comparing $3$ and $2$, we see that $3 > 2$.

Therefore, $\frac{3}{5} > \frac{2}{5}$.

This means Ritu ate more part of the apple than Somu.

To find by how much Ritu ate more, we subtract the part eaten by Somu from the part eaten by Ritu.

Difference $= \text{Part eaten by Ritu} - \text{Part eaten by Somu}

Difference $= \frac{3}{5} - \frac{2}{5}$

Subtract the numerators:

Difference $= \frac{3 - 2}{5} = \frac{1}{5}$

Ritu ate $\mathbf{\frac{1}{5}}$ part more than Somu.

Given:

Length of the wire $= 15.25$ cm.

The wire is bent to form a square.

To Find:

The length of each side of the square.

Solution:

When a wire is bent to form a square, the length of the wire is equal to the perimeter of the square.

Length of the wire = Perimeter of the square

Perimeter of the square $= 15.25$ cm.

The formula for the perimeter of a square is:

Perimeter $= 4 \times \text{side}$

Let the length of each side of the square be $s$ cm.

So, $4 \times s = 15.25$

To find the length of the side, we divide the perimeter by $4$:

$s = \frac{15.25}{4}$

Now, perform the division $15.25 \div 4$:

$\begin{array}{r} 3.8125\phantom{)} \\ 4{\overline{\smash{\big)}\,15.2500\phantom{)}}} \\ \underline{-~\phantom{(}(12)}\phantom{2500} \\ 32\phantom{500} \\ \underline{-~\phantom{()}(32)}\phantom{500} \\ 05\phantom{00} \\ \underline{-~\phantom{()}(4)}\phantom{00} \\ 10\phantom{0} \\ \underline{-~\phantom{()}(8)}\phantom{0} \\ 20\phantom{)} \\ \underline{-~\phantom{()}(20)} \\ 0\phantom{)} \end{array}$

So, $s = 3.8125$ cm.

The length of each side of the square will be $\mathbf{3.8125}$ cm.