Chapter 2 Fractions and Decimals (Concepts)

Fraction & Its Classification

In earlier classes, we worked with whole numbers and integers. Now, in Class 7, we will build upon our understanding of fractions and decimals, which are essential for representing parts of a whole or a collection. We will revisit the definition and types of fractions and decimals and then learn how to perform arithmetic operations with them.

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What is a Fraction?

A Fraction is a number that represents a part of a whole or a part of a collection. For a quantity to be represented as a fraction, the whole (object or collection) must be divided into a number of equal parts. A fraction tells us how many of these equal parts are being considered.

A fraction is written with two numbers separated by a horizontal line. The number above the line is called the numerator, and the number below the line is called the denominator.

• The Numerator (the number above the fraction bar) indicates the number of equal parts of the whole that are being taken or counted.
• The Denominator (the number below the fraction bar) represents the total number of equal parts into which the whole has been divided. The denominator can never be zero, as division by zero is undefined.

For example, if a pizza is cut into 8 equal slices and you eat 3 of these slices, the fraction representing the part of the pizza you ate is $\frac{3}{8}$. Here, 3 is the numerator (parts eaten), and 8 is the denominator (total equal parts).

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Classification of Fractions

Fractions are classified into different types based on the relationship between their numerator and their denominator. This classification helps us understand the value of a fraction relative to a whole number, especially 1.

1. Proper Fractions

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

Proper fractions always represent a quantity that is less than one whole unit. This makes sense because you are considering fewer parts than the total number of parts the whole was divided into.

Examples: $\frac{1}{2}$, $\frac{3}{4}$, $\frac{5}{7}$, $\frac{99}{100}$, $\frac{2}{5}$. In the fraction $\frac{3}{4}$, the numerator 3 is less than the denominator 4. If you take 3 slices from a cake cut into 4 equal slices, you have less than the whole cake.

2. Improper Fractions

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

Improper fractions represent a quantity that is equal to or greater than one whole unit. This happens when you have taken enough parts to make up one whole or even more than one whole.

Examples: $\frac{5}{3}$, $\frac{7}{4}$, $\frac{8}{8}$, $\frac{10}{3}$, $\frac{100}{27}$.

• For $\frac{5}{3}$, the numerator 5 is greater than the denominator 3. This means you have more than one whole.
• For $\frac{8}{8}$, the numerator 8 is equal to the denominator 8. This means you have taken all 8 parts from a whole divided into 8 parts, which is exactly 1 whole unit. ($\frac{8}{8} = 1$). Fractions equal to 1 are considered improper fractions.

3. Mixed Fractions (or Mixed Numbers)

A Mixed Fraction (or mixed number) is a way of writing improper fractions as a combination of a whole number and a proper fraction. Mixed fractions are always greater than 1 (as they contain a whole number part which is typically 1 or more). They provide a more intuitive way to express improper fractions in terms of wholes and remaining parts.

Example: $2\frac{1}{3}$. This is read as "two and one-third". It represents $2$ whole units plus $\frac{1}{3}$ of another unit.

Mixed fractions and improper fractions represent the same quantity, just in different forms. We can convert between them.

Converting an Improper Fraction to a Mixed Fraction:

To convert an improper fraction $\frac{N}{D}$ (where $N \ge D$) to a mixed fraction, divide the numerator ($N$) by the denominator ($D$).

• The result of the division gives a Quotient and a Remainder.
• The Quotient becomes the whole number part of the mixed fraction.
• The Remainder becomes the numerator of the proper fraction part.
• The original Denominator ($D$) remains the denominator of the proper fraction part.

Example: Convert the improper fraction $\frac{7}{4}$ to a mixed fraction.

Divide the numerator 7 by the denominator 4:

From the long division, we get:

• Quotient = 1
• Remainder = 3
• Divisor (original denominator) = 4

So, $\frac{7}{4}$ as a mixed fraction is $1\frac{3}{4}$.

Converting a Mixed Fraction to an Improper Fraction:

To convert a mixed fraction $W\frac{P}{D}$ (where $W$ is the whole number part, $P$ is the numerator, and $D$ is the denominator of the proper fraction part) to an improper fraction, follow these steps:

1. Multiply the whole number part ($W$) by the denominator ($D$).
2. Add the numerator of the proper fraction ($P$) to the result from step 1. This gives the numerator of the improper fraction.
3. The denominator of the improper fraction is the same as the original denominator ($D$).

Example: Convert the mixed fraction $2\frac{1}{3}$ to an improper fraction.

Here, $W=2, P=1, D=3$.

Multiply whole number by denominator: $2 \times 3 = 6$.

Add the numerator: $6 + 1 = 7$. This is the new numerator.

Keep the original denominator: $3$.

Classification of Fractions (Based on Denominators - for Comparison/Operations)

When dealing with groups of fractions, particularly for addition, subtraction, or comparison, we often classify them based on whether their denominators are the same or different.

• Like Fractions: Fractions that share the same denominator are called like fractions. They represent parts of wholes that have been divided into the same number of equal pieces, making comparison and operations simpler.

  Examples: $\frac{1}{7}$, $\frac{3}{7}$, $\frac{5}{7}$, $\frac{6}{7}$. All these fractions have the denominator 7.

  Other examples: $\frac{2}{15}, \frac{8}{15}, \frac{11}{15}$.

• Unlike Fractions: Fractions that have different denominators are called unlike fractions. They represent parts of wholes that have been divided into varying numbers of equal pieces, making direct comparison or addition/subtraction more involved.

  Examples: $\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{8}$, $\frac{4}{7}$. Their denominators (3, 5, 8, 7) are all different.

  Other examples: $\frac{1}{2}, \frac{2}{9}, \frac{5}{6}$.

4. Equivalent Fractions

Equivalent Fractions are fractions that look different (have different numerators and denominators) but represent the exact same value or the same part of a whole. They are equal in value.

You can obtain an equivalent fraction from a given fraction by multiplying or dividing both the numerator and the denominator by the same non-zero number. This is like multiplying or dividing the fraction by 1 (since $\frac{m}{m} = 1$ for $m \neq 0$), which does not change its value.

Example: Consider $\frac{1}{2}$. Multiplying numerator and denominator by 2 gives $\frac{1 \times 2}{2 \times 2} = \frac{2}{4}$. Multiplying by 3 gives $\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$. Multiplying by 5 gives $\frac{1 \times 5}{2 \times 5} = \frac{5}{10}$. So, $\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{5}{10}$ are all equivalent fractions. They all represent half of a whole.

Checking for Equivalent Fractions

To check if two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent, we can use the method of cross-multiplication. The fractions are equivalent if the product of the numerator of the first fraction and the denominator of the second fraction ($a \times d$) is equal to the product of the denominator of the first fraction and the numerator of the second fraction ($b \times c$).

Example: Are $\frac{2}{3}$ and $\frac{4}{6}$ equivalent?

Cross-multiply: Calculate $a \times d = 2 \times 6 = 12$.

Calculate $b \times c = 3 \times 4 = 12$.

Since $12 = 12$, the fractions $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent.

5. Unit Fractions

A Unit Fraction is a fraction whose numerator is 1. It represents a single one of the equal parts into which the whole has been divided.

Examples: $\frac{1}{2}$, $\frac{1}{5}$, $\frac{1}{10}$, $\frac{1}{23}$, $\frac{1}{100}$.

Any fraction $\frac{a}{b}$ can be thought of as '$a$' number of unit fractions $\frac{1}{b}$. For example, $\frac{3}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5}$.

6. Simplest Form of a Fraction (or Lowest Terms)

A fraction $\frac{a}{b}$ is said to be in its Simplest Form or Lowest Terms if the only common factor between its numerator ('a') and its denominator ('b') is 1. This means their Highest Common Factor (HCF) is 1.

To reduce a fraction to its simplest form, divide both the numerator and the denominator by their HCF.

Example: Reduce the fraction $\frac{12}{18}$ to its simplest form.

Find the factors of 12 and 18:

Factors of 12: 1, 2, 3, 4, 6, 12

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

The common factors are 1, 2, 3, and 6. The Highest Common Factor (HCF) is 6.

Divide both the numerator and the denominator of $\frac{12}{18}$ by their HCF (6):

So, the simplest form of $\frac{12}{18}$ is $\frac{2}{3}$. The HCF of 2 and 3 is 1.

Alternatively, you can use cancellation by repeatedly dividing the numerator and denominator by common factors:

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Comparison of Fractions

Comparing fractions means determining which fraction has a greater or smaller value, or if they are equal. The method for comparing fractions depends on whether they are like fractions or unlike fractions.

1. Comparing Like Fractions:

If two or more fractions have the same denominator (like fractions), it means the wholes are divided into the same number of equal parts. To compare them, you just need to compare the number of parts being considered (the numerators).

Rule: When comparing two or more like fractions, the fraction with the greater numerator is the greater fraction.

Example: Compare $\frac{3}{7}$ and $\frac{5}{7}$.

Both fractions have the same denominator, 7. Compare the numerators: 3 and 5. Since $5 > 3$, the fraction $\frac{5}{7}$ is greater than $\frac{3}{7}$.

2. Comparing Unlike Fractions with the Same Numerator:

If two fractions have the same numerator but different denominators, it means you are considering the same number of parts, but the size of the parts is different. A smaller denominator means the whole was divided into fewer parts, so each part is larger.

Rule: If two fractions have the same numerator, the fraction with the smaller denominator is the greater fraction.

Example: Compare $\frac{2}{5}$ and $\frac{2}{7}$.

Both fractions have the same numerator, 2. The denominators are 5 and 7. Since the denominator $5 < 7$, the fraction $\frac{2}{5}$ (where the whole is divided into 5 parts) is greater than $\frac{2}{7}$ (where the whole is divided into 7 parts).

Think of dividing a cake into 5 equal slices versus dividing the same cake into 7 equal slices. A slice from the 5-slice cake is bigger than a slice from the 7-slice cake.

3. Comparing Unlike Fractions with Different Numerators and Denominators:

When fractions have both different numerators and different denominators, we cannot compare them directly. We need to convert them into equivalent fractions that have the same denominator (like fractions).

Method 1: Convert to Like Fractions (using LCM)

1. Find the Least Common Multiple (LCM) of the denominators of the unlike fractions. This LCM will be the common denominator for the equivalent fractions.
2. Convert each unlike fraction into an equivalent fraction with the LCM as the denominator. Do this by multiplying both the numerator and the denominator of each fraction by the factor needed to change the original denominator to the LCM.
3. Now that you have like fractions, compare their numerators. The fraction with the greater numerator is the greater fraction.

Example: Compare $\frac{2}{3}$ and $\frac{3}{4}$.

These are unlike fractions with denominators 3 and 4.

Find the LCM of 3 and 4:

LCM$(3, 4) = 2 \times 2 \times 3 = 12$. The common denominator will be 12.

Convert $\frac{2}{3}$ to an equivalent fraction with denominator 12. Multiply numerator and denominator by 4 (since $3 \times 4 = 12$).

Convert $\frac{3}{4}$ to an equivalent fraction with denominator 12. Multiply numerator and denominator by 3 (since $4 \times 3 = 12$).

Now compare the like fractions $\frac{8}{12}$ and $\frac{9}{12}$. Compare their numerators: $8$ and $9$. Since $9 > 8$, $\frac{9}{12} > \frac{8}{12}$.

Therefore, $\frac{3}{4} > \frac{2}{3}$.

Method 2: Cross-Multiplication

This method allows you to compare two fractions without finding the LCM, using the principle related to equivalent fractions.

To compare $\frac{a}{b}$ and $\frac{c}{d}$:

Multiply the numerator of the first fraction by the denominator of the second: $a \times d$.

Multiply the denominator of the first fraction by the numerator of the second: $b \times c$.

Compare the two products:

• If $a \times d > b \times c$, then $\frac{a}{b} > \frac{c}{d}$.
• If $a \times d < b \times c$, then $\frac{a}{b} < \frac{c}{d}$.
• If $a \times d = b \times c$, then $\frac{a}{b} = \frac{c}{d}$ (they are equivalent).

Example: Compare $\frac{2}{3}$ and $\frac{3}{4}$ using cross-multiplication.

Here, $a=2, b=3, c=3, d=4$.

Calculate $a \times d$: $2 \times 4 = 8$.

Calculate $b \times c$: $3 \times 3 = 9$.

Compare the products: $8$ and $9$. Since $8 < 9$, according to the rule, $\frac{a}{b} < \frac{c}{d}$.

This method is fast for comparing two fractions, but for ordering more than two fractions, converting to like fractions (Method 1) is usually easier.

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Addition and Subtraction of Fractions

Understanding how to combine fractions through addition and subtraction is a fundamental skill. The approach we take depends on whether the fractions share the same denominator (like fractions) or have different denominators (unlike fractions). Performing these operations allows us to solve problems involving combining or finding the difference between parts of a whole.

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1. Addition and Subtraction of Like Fractions

Like Fractions are fractions that have the same denominator. Adding or subtracting like fractions is straightforward because the 'units' (the size of the equal parts) are the same. We simply combine or find the difference between the number of parts (the numerators).

Rule: To add or subtract like fractions, add or subtract their numerators and keep the common denominator the same.

Let $\frac{a}{c}$ and $\frac{b}{c}$ be two like fractions, where $c \neq 0$.

Formula for Addition:

Formula for Subtraction:

Conceptual Derivation:

Imagine a pizza cut into $c$ equal slices. The fraction $\frac{a}{c}$ represents $a$ of these slices. The fraction $\frac{b}{c}$ represents $b$ of these same-sized slices. When you add $\frac{a}{c}$ and $\frac{b}{c}$, you are combining the number of slices. You have a total of $(a+b)$ slices, and each slice is still $\frac{1}{c}$ of the whole pizza. So, the total is $\frac{a+b}{c}$.

Similarly, when you subtract $\frac{b}{c}$ from $\frac{a}{c}$, you are finding the difference in the number of slices. If you start with $a$ slices and take away $b$ slices, you are left with $(a-b)$ slices. Since each slice is still $\frac{1}{c}$ of the whole, the difference is $\frac{a-b}{c}$.

Examples:

(a) Add $\frac{2}{7}$ and $\frac{3}{7}$.

These are like fractions with denominator 7. Add the numerators and keep the denominator:

$\frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7}$.

(b) Subtract $\frac{2}{9}$ from $\frac{7}{9}$.

This means calculate $\frac{7}{9} - \frac{2}{9}$. These are like fractions with denominator 9. Subtract the numerators and keep the denominator:

$\frac{7}{9} - \frac{2}{9} = \frac{7-2}{9} = \frac{5}{9}$.

(c) Solve: $\frac{5}{8} + \frac{1}{8} - \frac{3}{8}$.

These are all like fractions with denominator 8. Perform the operations on the numerators from left to right, keeping the common denominator:

$\frac{5}{8} + \frac{1}{8} - \frac{3}{8} = \frac{5+1-3}{8} = \frac{6-3}{8} = \frac{3}{8}$.

Always check if the resulting fraction can be simplified to its lowest terms. In these examples, $\frac{5}{7}, \frac{5}{9},$ and $\frac{3}{8}$ are already in their simplest forms.

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2. Addition and Subtraction of Unlike Fractions

Unlike Fractions have different denominators. To add or subtract unlike fractions, we cannot directly combine their numerators because the sizes of the parts are different. We first need to convert them into equivalent like fractions. This means finding equivalent fractions for each of the original fractions such that they all share a common denominator. The easiest common denominator to use is the Least Common Multiple (LCM) of the original denominators.

Steps to Add or Subtract Unlike Fractions:

1. Find the Least Common Multiple (LCM) of the denominators of all the given unlike fractions. This LCM will be the common denominator for the equivalent fractions.
2. Convert each unlike fraction into an equivalent fraction that has the LCM as its denominator. To do this for each fraction, divide the LCM by the original denominator to find the necessary multiplying factor. Then, multiply both the numerator and the denominator of that fraction by this factor.
3. Once all the fractions have been converted into equivalent like fractions, add or subtract their numerators following the rules for adding or subtracting like fractions (as explained in the previous section).
4. Keep the common denominator (which is the LCM you found in Step 1).
5. Simplify the resulting fraction to its lowest terms if possible.

Example (Addition): Add $\frac{1}{4}$ and $\frac{2}{3}$.

These are unlike fractions (denominators 4 and 3).

Step 1: Find the LCM of the denominators 4 and 3. The multiples of 4 are 4, 8, 12, 16, ... The multiples of 3 are 3, 6, 9, 12, 15, ... The smallest common multiple is 12. So, LCM(4, 3) = 12.

Step 2: Convert $\frac{1}{4}$ and $\frac{2}{3}$ into equivalent fractions with a denominator of 12.

• For $\frac{1}{4}$: To get a denominator of 12, we multiply 4 by 3 ($12 \div 4 = 3$). So, multiply the numerator and denominator by 3: $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$.
• For $\frac{2}{3}$: To get a denominator of 12, we multiply 3 by 4 ($12 \div 3 = 4$). So, multiply the numerator and denominator by 4: $\frac{2 \times 4}{3 \times 4} = \frac{8}{12}$.

Step 3 & 4: Now add the equivalent like fractions $\frac{3}{12}$ and $\frac{8}{12}$. Add numerators ($3+8=11$) and keep the denominator (12).

$\frac{3}{12} + \frac{8}{12} = \frac{3+8}{12} = \frac{11}{12}$.

Step 5: Simplify $\frac{11}{12}$. The only common factor of 11 and 12 is 1. It is in simplest form.

So, $\frac{1}{4} + \frac{2}{3} = \frac{11}{12}$.

Example (Subtraction): Subtract $\frac{1}{6}$ from $\frac{3}{8}$. (i.e., calculate $\frac{3}{8} - \frac{1}{6}$)

These are unlike fractions (denominators 8 and 6).

Step 1: Find the LCM of the denominators 8 and 6.

LCM$(8, 6) = 2 \times 2 \times 2 \times 3 = 24$. The common denominator will be 24.

Step 2: Convert $\frac{3}{8}$ and $\frac{1}{6}$ into equivalent fractions with a denominator of 24.

• For $\frac{3}{8}$: To get a denominator of 24, multiply 8 by 3 ($24 \div 8 = 3$). So, $\frac{3 \times 3}{8 \times 3} = \frac{9}{24}$.
• For $\frac{1}{6}$: To get a denominator of 24, multiply 6 by 4 ($24 \div 6 = 4$). So, $\frac{1 \times 4}{6 \times 4} = \frac{4}{24}$.

Step 3 & 4: Now subtract the numerators of the equivalent like fractions $\frac{9}{24}$ and $\frac{4}{24}$. Subtract numerators ($9-4=5$) and keep the denominator (24).

$\frac{9}{24} - \frac{4}{24} = \frac{9-4}{24} = \frac{5}{24}$.

Step 5: Simplify $\frac{5}{24}$. The only common factor of 5 and 24 is 1. It is in simplest form.

So, $\frac{3}{8} - \frac{1}{6} = \frac{5}{24}$.

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3. Addition and Subtraction of Mixed Fractions

Mixed fractions involve both a whole number part and a fractional part. To add or subtract mixed fractions, you have two main options:

Method 1: Convert to Improper Fractions

This is often the most straightforward method, especially for subtraction where borrowing might be needed.

1. Convert all mixed fractions in the expression into their equivalent improper fractions using the rule $W\frac{P}{D} = \frac{(W \times D) + P}{D}$.
2. Perform the addition or subtraction of these improper fractions using the rules for adding/subtracting like or unlike fractions (find a common denominator if needed).
3. Simplify the resulting fraction if possible. If the result is an improper fraction and the question requires a mixed number answer, convert the final result back to a mixed fraction by dividing the numerator by the denominator.

Example (Addition): Add $2\frac{1}{3}$ and $1\frac{1}{4}$.

Step 1: Convert to improper fractions.

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

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

Step 2: Add the improper fractions: $\frac{7}{3} + \frac{5}{4}$. Denominators are 3 and 4. LCM(3, 4) = 12.

• Convert $\frac{7}{3}$ to denominator 12: $\frac{7 \times 4}{3 \times 4} = \frac{28}{12}$.
• Convert $\frac{5}{4}$ to denominator 12: $\frac{5 \times 3}{4 \times 3} = \frac{15}{12}$.

Add the like fractions: $\frac{28}{12} + \frac{15}{12} = \frac{28+15}{12} = \frac{43}{12}$.

Step 3: Simplify $\frac{43}{12}$. It's an improper fraction. Convert it back to a mixed number by dividing 43 by 12.

Quotient = 3, Remainder = 7, Denominator = 12. So, $\frac{43}{12} = 3\frac{7}{12}$.

Thus, $2\frac{1}{3} + 1\frac{1}{4} = 3\frac{7}{12}$.

Method 2: Add/Subtract Whole Parts and Fractional Parts Separately

This method can sometimes be quicker for addition, but requires careful handling of borrowing in subtraction.

1. Separate the whole number parts and the fractional parts of the mixed fractions.
2. Add or subtract the whole number parts separately.
3. Add or subtract the fractional parts separately (convert them to like fractions first if necessary).
4. Combine the results from step 2 and 3. If the resulting fractional part is an improper fraction, convert it to a mixed number and add its whole part to the sum of the original whole numbers. If performing subtraction of fractional parts requires borrowing, borrow 1 from the whole number part of the minuend and add it to its fractional part before subtracting.

Example (Addition using Method 2): Add $2\frac{1}{3}$ and $1\frac{1}{4}$.

$2\frac{1}{3} = 2 + \frac{1}{3}$. $1\frac{1}{4} = 1 + \frac{1}{4}$.

Add whole parts: $2 + 1 = 3$.

Add fractional parts: $\frac{1}{3} + \frac{1}{4}$. Denominators 3, 4. LCM(3, 4) = 12.

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

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

Sum of fractional parts $= \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$.

Combine results: Whole sum + Fractional sum $= 3 + \frac{7}{12} = 3\frac{7}{12}$.

Example (Subtraction using Method 2, with borrowing): Subtract $1\frac{3}{4}$ from $3\frac{1}{2}$. (i.e., $3\frac{1}{2} - 1\frac{3}{4}$)

The fractional parts are $\frac{1}{2}$ and $\frac{3}{4}$. LCM(2, 4) = 4.

Convert to like fractions: $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

So we need to calculate $3\frac{2}{4} - 1\frac{3}{4}$.

Notice that the first fractional part $\frac{2}{4}$ is smaller than the second fractional part $\frac{3}{4}$. We need to borrow from the whole number part of $3\frac{2}{4}$.

Borrow 1 from the whole number 3. The 3 becomes 2. The 1 borrowed is equivalent to $\frac{4}{4}$ (since the denominator is 4). Add this $\frac{4}{4}$ to the fractional part $\frac{2}{4}$.

$3\frac{2}{4} = (2 + 1) + \frac{2}{4} = 2 + \frac{4}{4} + \frac{2}{4} = 2 + \frac{4+2}{4} = 2\frac{6}{4}$.

Now subtract the mixed numbers: $2\frac{6}{4} - 1\frac{3}{4}$.

Subtract whole parts: $2 - 1 = 1$.

Subtract fractional parts: $\frac{6}{4} - \frac{3}{4} = \frac{6-3}{4} = \frac{3}{4}$.

Combine the results: $1 + \frac{3}{4} = 1\frac{3}{4}$.

Thus, $3\frac{1}{2} - 1\frac{3}{4} = 1\frac{3}{4}$. As seen here, the improper fraction method (Method 1) can sometimes be simpler for subtraction.

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Properties of Addition of Fractions

Addition of fractions follows the same basic properties as addition of whole numbers and integers:

Let $\frac{a}{b}$, $\frac{c}{d}$, and $\frac{e}{f}$ be any three fractions.

1. Closure Property: The sum of any two fractions is always a fraction. When you add two fractions, the result is a number that can also be expressed as a fraction.

   $\frac{a}{b} + \frac{c}{d}$ is always a fraction.

2. Commutative Property: Addition of fractions is commutative. This means that changing the order in which you add two fractions does not change the sum.

   $\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}$.

   Example: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$. And $\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$.

3. Associative Property: Addition of fractions is associative. When adding three or more fractions, the way you group them (which two you add first) does not affect the final sum.

   $(\frac{a}{b} + \frac{c}{d}) + \frac{e}{f} = \frac{a}{b} + (\frac{c}{d} + \frac{e}{f})$.

   Example: $(\frac{1}{4} + \frac{1}{2}) + \frac{1}{3} = (\frac{1}{4} + \frac{2}{4}) + \frac{1}{3} = \frac{3}{4} + \frac{1}{3} = \frac{9}{12} + \frac{4}{12} = \frac{13}{12}$.

   $\frac{1}{4} + (\frac{1}{2} + \frac{1}{3}) = \frac{1}{4} + (\frac{3}{6} + \frac{2}{6}) = \frac{1}{4} + \frac{5}{6} = \frac{3}{12} + \frac{10}{12} = \frac{13}{12}$.

   Both ways give the same result.

4. Additive Identity: Zero is the additive identity for fractions. The sum of any fraction and zero is the fraction itself. Zero can be written as a fraction $\frac{0}{b}$ where $b \neq 0$.

   $\frac{a}{b} + 0 = 0 + \frac{a}{b} = \frac{a}{b}$.

   Example: $\frac{3}{5} + 0 = \frac{3}{5}$.

Note: Subtraction of fractions, like subtraction of integers, is not commutative (e.g., $\frac{1}{2} - \frac{1}{3} \neq \frac{1}{3} - \frac{1}{2}$) and not associative (e.g., $(\frac{3}{4} - \frac{1}{2}) - \frac{1}{8} \neq \frac{3}{4} - (\frac{1}{2} - \frac{1}{8})$).

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Multiplication of Fractions

Multiplication of fractions allows us to find a part of a part, or to calculate the total amount when a fraction represents a quantity per unit. It is a fundamental operation in working with fractions and has many real-world applications. Unlike addition and subtraction, multiplication of fractions does not require converting them to a common denominator.

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1. Multiplication of a Fraction by a Whole Number

Multiplying a fraction by a whole number is equivalent to repeated addition of the fraction. For example, $3 \times \frac{2}{5}$ means $\frac{2}{5} + \frac{2}{5} + \frac{2}{5}$.

Rule: To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction, and keep the denominator the same.

We can think of a whole number '$w$' as a fraction $\frac{w}{1}$. Then we apply the rule for multiplying two fractions.

Formula:

Example: Multiply $3 \times \frac{2}{5}$.

Using the formula:

The result is the improper fraction $\frac{6}{5}$. We can convert it to a mixed fraction by dividing 6 by 5 ($6 \div 5 = 1$ with remainder 1). So, $\frac{6}{5} = 1\frac{1}{5}$.

Using repeated addition confirms this: $\frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{2+2+2}{5} = \frac{6}{5}$.

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2. Multiplication of a Fraction by a Fraction

Multiplying a fraction by another fraction is like finding a fraction 'of' another fraction (e.g., $\frac{1}{2} \times \frac{1}{4}$ means half of one-fourth). This operation is straightforward.

Rule: To multiply two or more fractions, multiply all the numerators together to get the new numerator, and multiply all the denominators together to get the new denominator. Simplify the resulting fraction if possible.

Let $\frac{a}{b}$ and $\frac{c}{d}$ be two fractions (where $b \neq 0, d \neq 0$).

Formula:

Example: Multiply $\frac{2}{3} \times \frac{4}{5}$.

Multiply numerators: $2 \times 4 = 8$. Multiply denominators: $3 \times 5 = 15$.

The fraction $\frac{8}{15}$ is in simplest form (HCF of 8 and 15 is 1).

Simplifying Before Multiplication (Cancellation)

To make multiplication easier and avoid simplifying large numbers at the end, you can cancel out common factors between any numerator and any denominator *before* performing the multiplication. This is valid because you are essentially dividing both a numerator and a denominator by the same number, which is allowed in multiplication of fractions.

Example with cancellation: Multiply $\frac{3}{8} \times \frac{4}{9}$.

We can write this as $\frac{3 \times 4}{8 \times 9}$.

Look for common factors between a numerator and a denominator (it can be across the fractions):

• Numerator 3 and Denominator 9 have a common factor of 3. Divide 3 by 3 to get 1, and 9 by 3 to get 3.
• Numerator 4 and Denominator 8 have a common factor of 4. Divide 4 by 4 to get 1, and 8 by 4 to get 2.

Using cancellation notation:

Now the problem becomes $\frac{1}{8} \times \frac{4}{3}$. We can see that numerator 4 and denominator 8 still have a common factor of 4 (or use the factors 2 from the previous step).

Now multiply the remaining terms:

Without cancellation first: $\frac{3 \times 4}{8 \times 9} = \frac{12}{72}$. To simplify $\frac{12}{72}$, find the HCF of 12 and 72. HCF(12, 72) = 12. $\frac{12 \div 12}{72 \div 12} = \frac{1}{6}$. Both methods give the same correct result, but cancellation before multiplying is usually faster.

The word "of" means Multiplication:

In problems involving fractions, the word "of" between a quantity and a fraction indicates the operation of multiplication. For example, "$\frac{1}{2}$ of $10$ oranges" means $\frac{1}{2} \times 10$ oranges.

Example: Find $\frac{2}{5}$ of $25 \textsf{₹}$.

This translates to $\frac{2}{5} \times 25$. Remember to write the whole number 25 as a fraction $\frac{25}{1}$.

Multiply numerators and denominators, or use cancellation:

So, $\frac{2}{5}$ of $25 \textsf{₹}$ is $10 \textsf{₹}$.

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3. Multiplication of Mixed Fractions

To multiply mixed fractions, the easiest approach is to first convert them into their equivalent improper fractions. Then, multiply the improper fractions using the rules for multiplying fractions.

Steps to Multiply Mixed Fractions:

1. Convert each mixed fraction into an improper fraction.
2. Multiply the improper fractions as described in the previous section (multiply numerators and denominators, simplifying by cancellation if possible).
3. If the result is an improper fraction, convert it back to a mixed fraction (or simplify it if it's not in lowest terms).

Example: Multiply $2\frac{1}{3} \times 1\frac{1}{2}$.

Step 1: Convert to improper fractions.

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

Step 2: Multiply the improper fractions $\frac{7}{3} \times \frac{3}{2}$.

Using cancellation:

Step 3: The result is the improper fraction $\frac{7}{2}$. Convert it to a mixed fraction by dividing 7 by 2 ($7 \div 2 = 3$ with remainder 1).

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

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Properties of Multiplication of Fractions

Multiplication of fractions, like addition, follows certain fundamental properties:

Let $\frac{a}{b}$, $\frac{c}{d}$, and $\frac{e}{f}$ be any three fractions.

1. Closure Property: The product of any two fractions is always a fraction. The set of fractions is closed under multiplication.

   $\frac{a}{b} \times \frac{c}{d}$ is always a fraction.

   Example: $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$, which is a fraction.

2. Commutative Property: Multiplication of fractions is commutative. The order in which you multiply two fractions does not change the product.

   $\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}$.

   Example: $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$. And $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$.

3. Associative Property: Multiplication of fractions is associative. When multiplying three or more fractions, the way you group them using parentheses does not affect the final product.

   $(\frac{a}{b} \times \frac{c}{d}) \times \frac{e}{f} = \frac{a}{b} \times (\frac{c}{d} \times \frac{e}{f})$.

   Example: $(\frac{1}{2} \times \frac{1}{3}) \times \frac{1}{4} = \frac{1}{6} \times \frac{1}{4} = \frac{1}{24}$.

   $\frac{1}{2} \times (\frac{1}{3} \times \frac{1}{4}) = \frac{1}{2} \times \frac{1}{12} = \frac{1}{24}$. Both ways give the same result.

4. Multiplicative Identity: The number 1 is the multiplicative identity for fractions. When any fraction is multiplied by 1, its value does not change. The number 1 can be written as $\frac{1}{1}$ or any equivalent fraction like $\frac{2}{2}, \frac{3}{3}$, etc.

   $\frac{a}{b} \times 1 = 1 \times \frac{a}{b} = \frac{a}{b}$.

   Example: $\frac{3}{5} \times 1 = \frac{3}{5}$.

5. Multiplicative Property of Zero: The product of any fraction and zero is zero.

   $\frac{a}{b} \times 0 = 0 \times \frac{a}{b} = 0$.

   Example: $\frac{7}{8} \times 0 = 0$.

6. Distributive Property of Multiplication over Addition: Multiplication distributes over addition for fractions. This means $a \times (b + c) = a \times b + a \times c$ holds for fractions too.

   $\frac{a}{b} \times (\frac{c}{d} + \frac{e}{f}) = (\frac{a}{b} \times \frac{c}{d}) + (\frac{a}{b} \times \frac{e}{f})$.

7. Distributive Property of Multiplication over Subtraction: Multiplication also distributes over subtraction for fractions.

   $\frac{a}{b} \times (\frac{c}{d} - \frac{e}{f}) = (\frac{a}{b} \times \frac{c}{d}) - (\frac{a}{b} \times \frac{e}{f})$.

Answer:

Answer:

Division of Fractions

Division of fractions is the inverse operation of multiplication. This means that division "undoes" what multiplication does. For example, if $3 \times \frac{1}{2} = \frac{3}{2}$, then $\frac{3}{2} \div \frac{1}{2} = 3$ and $\frac{3}{2} \div 3 = \frac{1}{2}$. To understand how to divide fractions, we first need to learn about the concept of a reciprocal.

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Reciprocal of a Fraction

The Reciprocal of a non-zero fraction is obtained by simply flipping the fraction upside down, i.e., interchanging its numerator and its denominator. The reciprocal of a fraction is also known as its Multiplicative Inverse.

If $\frac{a}{b}$ is a non-zero fraction (meaning $a \neq 0$ and $b \neq 0$), its reciprocal is $\frac{b}{a}$.

A key property of reciprocals is that the product of any non-zero fraction and its reciprocal is always 1 (the multiplicative identity).

Examples of Reciprocals:

• The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. Check: $\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$.
• The reciprocal of $\frac{7}{1}$ (which is the whole number 7) is $\frac{1}{7}$. Check: $7 \times \frac{1}{7} = \frac{7}{7} = 1$.
• The reciprocal of $\frac{1}{5}$ is $\frac{5}{1}$ (or 5). Check: $\frac{1}{5} \times 5 = 1$.
• The reciprocal of a proper fraction (like $\frac{2}{3}$) is always an improper fraction (like $\frac{3}{2}$).
• The reciprocal of an improper fraction (other than 1, like $\frac{5}{4}$) is always a proper fraction (like $\frac{4}{5}$).
• The reciprocal of 1 (which is $\frac{1}{1}$) is 1 ($\frac{1}{1}$).
• Zero (0, which can be written as $\frac{0}{1}$) does not have a reciprocal, because if you tried to find its reciprocal, you would get $\frac{1}{0}$, which is undefined.

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Rule for Division of Fractions

The rule for dividing by a fraction is the core concept of fraction division:

Rule: To divide a quantity by a fraction, multiply the quantity by the reciprocal of the fraction you are dividing by (the divisor).

1. Division of a Whole Number by a Fraction

To divide a whole number '$w$' by a fraction $\frac{a}{b}$ ($a, b \neq 0$), multiply the whole number by the reciprocal of the fraction $\frac{a}{b}$ (which is $\frac{b}{a}$).

Formula:

Example: Divide $3 \div \frac{2}{5}$.

The whole number is 3. The fraction is $\frac{2}{5}$. The reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$.

Write 3 as $\frac{3}{1}$ and multiply:

The result is the improper fraction $\frac{15}{2}$. Convert to a mixed fraction by dividing 15 by 2 ($15 \div 2 = 7$ with remainder 1). So, $\frac{15}{2} = 7\frac{1}{2}$.

Interpretation: How many halves are there in 3? $3 \div \frac{1}{2} = 3 \times 2 = 6$. How many groups of $\frac{2}{5}$ are there in 3? The answer is $7\frac{1}{2}$.

2. Division of a Fraction by a Whole Number

To divide a fraction $\frac{a}{b}$ by a whole number '$w$' ($w \neq 0$), multiply the fraction by the reciprocal of the whole number. The reciprocal of a whole number $w$ (written as $\frac{w}{1}$) is $\frac{1}{w}$.

Formula:

Example: Divide $\frac{3}{4} \div 2$.

The fraction is $\frac{3}{4}$. The whole number (divisor) is 2. The reciprocal of 2 is $\frac{1}{2}$.

Multiply the fractions:

Interpretation: If you divide $\frac{3}{4}$ of a pizza equally among 2 people, each person gets $\frac{3}{8}$ of the pizza.

3. Division of a Fraction by Another Fraction

This is the most general case of fraction division. To divide one fraction by another non-zero fraction, multiply the first fraction (the dividend) by the reciprocal of the second fraction (the divisor).

Formula:

Example: Divide $\frac{2}{5} \div \frac{3}{7}$.

The first fraction (dividend) is $\frac{2}{5}$. The second fraction (divisor) is $\frac{3}{7}$.

Find the reciprocal of the divisor $\frac{3}{7}$: The reciprocal is $\frac{7}{3}$.

Now multiply the first fraction by the reciprocal of the second fraction:

Multiply the numerators and denominators:

The result is $\frac{14}{15}$. It is in simplest form.

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Division involving Mixed Fractions

To perform division when mixed fractions are involved, the first step is always to convert any mixed fractions into their equivalent improper fractions. Then, apply the rule for dividing fractions.

Steps to Divide Mixed Fractions:

1. Convert all mixed fractions into improper fractions.
2. Convert any whole numbers into fractions by writing them with a denominator of 1.
3. Rewrite the division problem with improper fractions (and fractions for whole numbers).
4. Perform the division using the rule: multiply the first fraction by the reciprocal of the second fraction (the divisor). Simplify by cancellation if possible.
5. If the resulting fraction is improper, convert it back to a mixed fraction (or simplify it if it's not in lowest terms).

Example: Divide $3\frac{1}{2} \div 1\frac{1}{4}$.

Step 1: Convert mixed fractions to improper fractions.

• $3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6+1}{2} = \frac{7}{2}$.
• $1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4+1}{4} = \frac{5}{4}$.

Step 2: Rewrite the division problem using the improper fractions: $\frac{7}{2} \div \frac{5}{4}$.

Step 3: Perform the division by multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of $\frac{5}{4}$ is $\frac{4}{5}$.

Multiply the fractions, using cancellation (2 and 4 have HCF 2):

Step 4: The result is the improper fraction $\frac{14}{5}$. Convert it to a mixed fraction by dividing 14 by 5 ($14 \div 5 = 2$ with a remainder of 4).

So, $\frac{14}{5} = 2\frac{4}{5}$.

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Properties related to Division of Fractions

Unlike multiplication, division of fractions does not follow the same set of properties in a general sense.

• Non-Closure: The set of integers is not closed under division (as seen in Chapter 1). The set of fractions (or rational numbers) *is* closed under division, but only if we exclude division by zero. The quotient of two fractions (where the divisor is not zero) is always a fraction.
• Not Commutative: Division of fractions is generally not commutative. $\frac{a}{b} \div \frac{c}{d} \neq \frac{c}{d} \div \frac{a}{b}$ unless the two fractions are equal in value (like $\frac{1}{2} \div \frac{1}{2} = 1$ and $\frac{1}{2} \div \frac{1}{2} = 1$).

Example: $\frac{1}{2} \div \frac{1}{4}$. Reciprocal of $\frac{1}{4}$ is $\frac{4}{1}$. $\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$.

Now swap the order: $\frac{1}{4} \div \frac{1}{2}$. Reciprocal of $\frac{1}{2}$ is $\frac{2}{1}$. $\frac{1}{4} \div \frac{1}{2} = \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2}$.

Since $2 \neq \frac{1}{2}$, division is not commutative.

• Not Associative: Division of fractions is also not associative. The way you group fractions when performing multiple divisions affects the result. $( \frac{a}{b} \div \frac{c}{d} ) \div \frac{e}{f} \neq \frac{a}{b} \div ( \frac{c}{d} \div \frac{e}{f} )$ generally.

Answer:

Introduction to Decimals

In our study of numbers, we first encountered whole numbers and then extended our understanding to include negative numbers (integers). We also learned about fractions, which are used to represent parts of a whole. Decimal Numbers, or simply decimals, are another way of writing fractions, particularly those whose denominators are powers of 10 (like 10, 100, 1000, etc.). Decimals provide a convenient and consistent way to work with these specific types of fractions, and they are widely used in everyday contexts such as money, measurements, and data analysis.

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What are Decimals?

A decimal number is a number that consists of two parts: a whole number part and a fractional part (or decimal part). These two parts are separated by a dot called the decimal point (.).

Example: Consider the decimal number $34.572$.

• The part to the left of the decimal point is the whole number part, which is $34$.
• The dot '.' is the decimal point.
• The part to the right of the decimal point is the decimal part, which consists of the digits $5, 7,$ and $2$.

The digits to the right of the decimal point represent fractional values where the denominators are powers of 10. For example, $0.1 = \frac{1}{10}$, $0.01 = \frac{1}{100}$, $0.001 = \frac{1}{1000}$, and so on.

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Place Value in Decimals

The place value system, which is fundamental to understanding the value of digits in whole numbers (ones, tens, hundreds, etc.), extends to the right of the decimal point to represent the fractional part. As we move one place to the right from the ones place, the value of the place becomes one-tenth of the previous place.

Let's look at the place values in a decimal number:

...	Thousands ($1000$)	Hundreds ($100$)	Tens ($10$)	Ones ($1$)	Decimal Point (.)	Tenths ($\frac{1}{10}$ or $0.1$)	Hundredths ($\frac{1}{100}$ or $0.01$)	Thousandths ($\frac{1}{1000}$ or $0.001$)	...
...	Digit in 1000s place	Digit in 100s place	Digit in 10s place	Digit in 1s place	.	Digit in 10ths place	Digit in 100ths place	Digit in 1000ths place	...
For 147.253		1	4	7	.	2	5	3

For the number $147.253$:

• The digit $1$ is in the Hundreds place. Its value is $1 \times 100 = 100$.
• The digit $4$ is in the Tens place. Its value is $4 \times 10 = 40$.
• The digit $7$ is in the Ones place. Its value is $7 \times 1 = 7$.
• The digit $2$ is in the Tenths place. Its value is $2 \times \frac{1}{10} = 0.2$.
• The digit $5$ is in the Hundredths place. Its value is $5 \times \frac{1}{100} = 0.05$.
• The digit $3$ is in the Thousandths place. Its value is $3 \times \frac{1}{1000} = 0.003$.

We can write a decimal number in expanded form by summing the place values of its digits:

Expanded form of $147.253 = (1 \times 100) + (4 \times 10) + \ $$ (7 \times 1) + (2 \times \frac{1}{10}) + \ $$ (5 \times \frac{1}{100}) + \ $$ (3 \times \frac{1}{1000})$.

Or, using decimal values for the fractional part: $100 + 40 + 7 + 0.2 + 0.05 + 0.003$.

Reading and Writing Decimals

To read a decimal number correctly:

1. Read the whole number part as usual.
2. Read the decimal point as "point".
3. Read each digit in the decimal part separately.

Example: $34.572$ is read as "thirty-four point five seven two". It is not read as "thirty-four point five hundred seventy-two".

Example: $0.6$ is read as "zero point six" or sometimes as "six tenths".

Example: $125.08$ is read as "one hundred twenty-five point zero eight".

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Converting Fractions to Decimals

Since decimals are a way to represent certain fractions, we can convert any fraction into its decimal form.

Method 1: By Making the Denominator a Power of 10

If the denominator of a fraction is already a power of 10 (10, 100, 1000, etc.) or can be easily converted into one by multiplying both the numerator and the denominator by the same suitable number, this method is quick.

After making the denominator a power of 10, write the numerator and place the decimal point. The number of digits after the decimal point must be equal to the number of zeros in the denominator. You might need to add zeros to the left of the numerator to achieve the correct number of decimal places.

Examples:

(a) Convert $\frac{3}{5}$ to a decimal.

The denominator is 5. We can make it 10 by multiplying by 2. Multiply both numerator and denominator by 2:

The denominator is 10 (one zero). Write the numerator 6 and place the decimal point 1 place from the right: $0.6$.

(b) Convert $\frac{7}{20}$ to a decimal.

The denominator is 20. We can make it 100 by multiplying by 5 ($20 \times 5 = 100$). Multiply both numerator and denominator by 5:

The denominator is 100 (two zeros). Write the numerator 35 and place the decimal point 2 places from the right: $0.35$.

(c) Convert $\frac{3}{8}$ to a decimal.

The denominator is 8. We can make it 1000 by multiplying by 125 ($8 \times 125 = 1000$). Multiply both numerator and denominator by 125:

The denominator is 1000 (three zeros). Write the numerator 375 and place the decimal point 3 places from the right: $0.375$.

Method 2: By Division

This method works for converting any fraction to a decimal, regardless of whether the denominator can be easily converted to a power of 10. It involves performing long division.

Rule: To convert a fraction to a decimal, divide the numerator by the denominator. Add a decimal point and zeros to the right of the numerator as needed to continue the division.

Example: Convert $\frac{3}{4}$ to a decimal using division.

Divide 3 by 4. Since 3 is less than 4, the whole number part of the quotient is 0. Add a decimal point after 3 and a zero to its right (3.0) and continue dividing.

The remainder is 0. This is a terminating decimal (the division ends).

Example: Convert $\frac{2}{3}$ to a decimal using division.

Divide 2 by 3. Since 2 is less than 3, start with 0, add a decimal point, and zeros.

The remainder 2 keeps repeating, and the digit 6 in the quotient keeps repeating. This is a non-terminating repeating (or recurring) decimal.

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Converting Decimals to Fractions

To convert a terminating decimal number back into a fraction, we use the place value of the last digit in the decimal part to determine the denominator.

Steps to convert a terminating decimal to a fraction:

1. Write the decimal number without the decimal point. This number becomes the numerator of the fraction.
2. Determine the denominator based on the number of digits after the decimal point in the original decimal number.
   • If there is 1 digit after the decimal, the denominator is 10.
   • If there are 2 digits after the decimal, the denominator is 100.
   • If there are 3 digits after the decimal, the denominator is 1000, and so on.
   • The denominator is $1$ followed by as many zeros as there are digits after the decimal point.
3. Form the fraction using the numerator from step 1 and the denominator from step 2.
4. Simplify the resulting fraction to its lowest terms by dividing the numerator and denominator by their HCF.

Examples:

(a) Convert $0.75$ to a fraction.

Step 1: Write the number without the decimal point: 75. This is the numerator.

Step 2: Count digits after the decimal point in $0.75$: There are 2 digits (7 and 5). So the denominator is 1 followed by two zeros: 100.

Step 3: Form the fraction: $\frac{75}{100}$.

Step 4: Simplify $\frac{75}{100}$ to lowest terms. HCF(75, 100) = 25. Divide numerator and denominator by 25.

(b) Convert $2.35$ to a fraction.

Method 1: Treat the whole number part separately.

$2.35 = 2 + 0.35$. Convert the decimal part $0.35$ to a fraction.

Numerator = 35. Decimal part has 2 digits, so denominator = 100. Fraction for 0.35 is $\frac{35}{100}$.

Simplify $\frac{35}{100}$. HCF(35, 100) = 5. $\frac{35 \div 5}{100 \div 5} = \frac{7}{20}$.

So, $2.35 = 2 + \frac{7}{20} = 2\frac{7}{20}$. (This is a mixed fraction).

Method 2: Convert directly to an improper fraction.

Step 1: Write the number without decimal point: 235. Numerator = 235.

Step 2: Count digits after decimal point in $2.35$: 2 digits. Denominator = 100.

Step 3: Fraction = $\frac{235}{100}$.

Step 4: Simplify $\frac{235}{100}$. HCF(235, 100) = 5. Divide by 5.

This is an improper fraction. Both results ($2\frac{7}{20}$ and $\frac{47}{20}$) are correct and equivalent.

(c) Convert $0.008$ to a fraction.

Step 1: Number without decimal point: 008, which is 8. Numerator = 8.

Step 2: Count digits after decimal point in $0.008$: 3 digits (0, 0, and 8). Denominator = 1000.

Step 3: Fraction = $\frac{8}{1000}$.

Step 4: Simplify $\frac{8}{1000}$. HCF(8, 1000) = 8. Divide by 8.

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Types of Decimals

Based on how the division of the numerator by the denominator behaves, decimals can be classified:

1. Terminating Decimals: These are decimal numbers where the division process ends, resulting in a finite number of digits after the decimal point. This happens when the prime factors of the denominator of the simplified fraction are only 2s and/or 5s.

   Examples: $0.5$ (from $\frac{1}{2}$, denominator $2^1$), $0.75$ (from $\frac{3}{4}$, denominator $2^2$), $0.125$ (from $\frac{1}{8}$, denominator $2^3$), $0.35$ (from $\frac{7}{20}$, denominator $2^2 \times 5^1$), $0.1$ (from $\frac{1}{10}$, denominator $2 \times 5$).

2. Non-Terminating Decimals: These are decimal numbers where the division process never ends, resulting in an infinite number of digits after the decimal point. This happens when the denominator of the simplified fraction has prime factors other than 2 or 5.

   • Non-Terminating Repeating (or Recurring) Decimals: A single digit or a block of digits repeats infinitely in a specific pattern. We indicate the repeating part by placing a bar over it.

   Examples: $0.333... = 0.\overline{3}$ (from $\frac{1}{3}$), $0.1666... = 0.1\overline{6}$ (from $\frac{1}{6}$, since $\frac{1}{6} = \frac{1}{2 \times 3}$), $0.142857142857... = 0.\overline{142857}$ (from $\frac{1}{7}$).

   • Non-Terminating Non-Repeating Decimals: The digits after the decimal point continue infinitely without any repeating pattern. These numbers are called irrational numbers (e.g., $\pi \approx 3.14159265...$, $\sqrt{2} \approx 1.41421356...$). Studying these is typically done in higher classes.

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Comparing Decimals

Comparing decimal numbers is similar to comparing whole numbers. We compare the digits from left to right, starting from the largest place value.

Steps to compare two decimal numbers:

1. Compare the whole number parts (the digits to the left of the decimal point). The decimal with the larger whole number part is the greater number.
2. If the whole number parts are equal, move to the right of the decimal point and compare the digits in the tenths place. The decimal with the larger digit in the tenths place is greater.
3. If the tenths digits are also equal, compare the digits in the hundredths place, and so on, moving one place to the right each time.
4. Continue this process until you find the first place value position where the digits are different. The decimal number with the larger digit in that position is the greater number.
5. (Optional but helpful): To make comparison easier, you can convert the unlike decimal numbers to like decimal numbers by adding zeros to the right end of the decimal part of the number(s) with fewer decimal places. Adding trailing zeros does not change the value of a decimal (e.g., $0.5 = 0.50 = 0.500$).

Like Decimals: Decimals having the same number of digits after the decimal point (e.g., 2.34, 0.56, 12.09 are like decimals with 2 decimal places).

Unlike Decimals: Decimals having a different number of digits after the decimal point (e.g., 2.3, 0.567, 12.0 are unlike decimals).

We can convert unlike decimals to like decimals by adding trailing zeros. Example: 2.3 (1 decimal place), 0.567 (3 decimal places), 12.0 (1 decimal place). The maximum is 3 decimal places. Convert them to like decimals: 2.300, 0.567, 12.000.

Examples of Comparing Decimals:

(a) Compare $3.5$ and $3.45$.

Compare whole parts: $3 = 3$.

Compare tenths digits: $5$ in $3.5$ and $4$ in $3.45$. Since $5 > 4$, $3.5 > 3.45$.

(Using like decimals: $3.50$ and $3.45$. Compare 3.50 and 3.45. $3=3$ (whole). $5>4$ (tenths). So, $3.50 > 3.45$.)

(b) Compare $0.07$ and $0.1$.

Compare whole parts: $0 = 0$.

Compare tenths digits: $0$ in $0.07$ and $1$ in $0.1$. Since $0 < 1$, $0.07 < 0.1$.

(Using like decimals: $0.07$ and $0.10$. Compare $0.07$ and $0.10$. $0=0$ (whole). $0<1$ (tenths). So, $0.07 < 0.10$.)

(c) Compare $5.608$ and $5.68$.

Compare whole parts: $5 = 5$.

Compare tenths digits: $6$ in $5.608$ and $6$ in $5.68$. Equal.

Compare hundredths digits: $0$ in $5.608$ and $8$ in $5.68$. Since $0 < 8$, $5.608 < 5.68$.

(Using like decimals: $5.608$ and $5.680$. Compare $5.608$ and $5.680$. $5=5$ (whole). $6=6$ (tenths). $0<8$ (hundredths). So, $5.608 < 5.680$.)

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Uses of Decimals

Decimal numbers are used extensively in everyday life and various fields because they are a convenient way to represent parts of units that are divided into tenths, hundredths, etc. They integrate seamlessly with our base-10 number system.

Here are some common uses:

• Money (Currency): Money is perhaps the most familiar use of decimals. In India, 1 Rupee = 100 paise. Paise are hundredths of a rupee. So, we write money using decimals:

  $1 \text{ paise} = \textsf{₹} \frac{1}{100} = \textsf{₹} 0.01 $

  Example: $\textsf{₹} 5.75$ means 5 rupees and 75 paise ($75 \text{ paise} = \textsf{₹} \frac{75}{100} = \textsf{₹} 0.75$).

  Example: $\textsf{₹} 12.50$ means 12 rupees and 50 paise ($50 \text{ paise} = \textsf{₹} \frac{50}{100} = \textsf{₹} 0.50$).

• Length Measurement: Metric units of length (millimetres, centimetres, metres, kilometres) are based on powers of 10, making decimals natural for expressing lengths.

  $1 \text{ cm} = \frac{1}{100} \text{ m} = 0.01 \text{ m} $

  $1 \text{ mm} = \frac{1}{10} \text{ cm} = 0.1 \text{ cm} $

  $1 \text{ m} = \frac{1}{1000} \text{ km} = 0.001 \text{ km} $

  Example: $2.5 \text{ m}$ means 2 metres and 5 tenths of a metre (which is $0.5 \times 100 \text{ cm} = 50 \text{ cm}$). So, $2.5 \text{ m} = 2 \text{ m } 50 \text{ cm}$.

  Example: $10.75 \text{ cm}$ means 10 cm and 75 hundredths of a cm (which is $0.75 \times 10 \text{ mm} = 7.5 \text{ mm}$). So $10.75 \text{ cm} = 10 \text{ cm } 7.5 \text{ mm}$.

• Weight (Mass) Measurement: Similarly, weights in kilograms and grams or grams and milligrams use decimals.

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

  Example: $3.250 \text{ kg}$ means 3 kilograms and 250 grams ($250 \text{ g} = \frac{250}{1000} \text{ kg} = 0.250 \text{ kg}$). So, $3.250 \text{ kg} = 3 \text{ kg } 250 \text{ g}$. Note that $0.250 \text{ kg}$ is the same as $0.25 \text{ kg}$.

• Capacity (Volume) Measurement: Volumes in litres and millilitres also use decimals.

  $1 \text{ mL} = \frac{1}{1000} \text{ L} = 0.001 \text{ L} $

  Example: $1.75 \text{ L}$ means 1 litre and 75 hundredths of a litre ($0.75 \times 1000 \text{ mL} = 750 \text{ mL}$). So, $1.75 \text{ L} = 1 \text{ L } 750 \text{ mL}$.

• Temperature: Temperature readings are often given in decimals (e.g., body temperature $37.5^\circ\text{C}$).
• Speed and Distance: Measurements like $55.2 \text{ km/h}$ or distances like $8.6 \text{ km}$.
• Reading Meters: Readings from electricity, water, and gas meters, or fuel pumps at petrol stations often show decimal values.

Working with decimals is often easier than working with fractions for addition, subtraction, and comparison, especially in practical measurements.

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Addition and Subtraction of Decimals

Performing addition and subtraction with decimal numbers is very similar to working with whole numbers. The most important rule is to align the decimal points of the numbers being added or subtracted. This ensures that you are adding or subtracting corresponding place values (tenths with tenths, hundredths with hundredths, and so on).

Addition of Decimals

To add decimal numbers, follow these steps:

1. Write the numbers one below the other in a column, making sure the decimal points are aligned.
2. (Recommended) Add trailing zeros to convert unlike decimals to like decimals. This helps in keeping the columns straight.
3. Add the numbers as you would add whole numbers, starting from the rightmost digit.
4. Place the decimal point in the sum directly below the decimal points in the numbers you added.

Example: Add $15.7$, $8.23$, and $0.542$.

Arrange the numbers vertically, aligning the decimal points and adding trailing zeros.

$$\begin{array}{cc} & 1 & 5 & . & 7 & 0 & 0 \\ & & 8 & . & 2 & 3 & 0 \\ + & & 0 & . & 5 & 4 & 2 \\ \hline & 2 & 4 & . & 4 & 7 & 2 \\ \hline \end{array}$$

The sum is $24.472$.

Subtraction of Decimals

To subtract decimal numbers, follow these steps:

1. Write the smaller number below the larger number, ensuring the decimal points are aligned.
2. It is essential to add trailing zeros to the number with fewer decimal places to make them like decimals. This is crucial for borrowing.
3. Subtract as you would subtract whole numbers, starting from the rightmost digit and borrowing when necessary.
4. Place the decimal point in the difference directly below the decimal points above.

Example: Subtract $6.78$ from $9.5$.

Arrange the numbers vertically. Add a zero to $9.5$ to make it $9.50$.

$$\begin{array}{cc} & 8 & & \stackrel{14}{\cancel{5}} & \stackrel{10}{\cancel{0}} \\ & 9 & . & \cancel{5} & \cancel{0} \\ - & 6 & . & 7 & 8 \\ \hline & 2 & . & 7 & 2 \\ \hline \end{array}$$

The difference is $2.72$.

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Multiplication and Division of Decimal Numbers

We have learned about decimals and their relationship with fractions. Now, we will perform the operations of multiplication and division using decimal numbers. These operations follow specific rules that ensure accuracy, and they are widely applied in practical problems involving measurements and quantities expressed as decimals.

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1. Multiplication of Decimal Numbers

Multiplying decimal numbers is similar to multiplying whole numbers, but with the added step of correctly placing the decimal point in the final product. The position of the decimal point depends on the total number of decimal places in the numbers being multiplied.

(a) Multiplication of a Decimal by a Whole Number

To multiply a decimal number by a whole number:

1. Ignore the decimal point in the decimal number and multiply the numbers as if they were both whole numbers.
2. Count the number of decimal places in the original decimal number (the number of digits to the right of the decimal point).
3. In the product obtained in Step 1, count the same number of places from the rightmost digit and place the decimal point there.

Example: Multiply $2.35 \times 4$.

Step 1: Multiply 235 by 4 as whole numbers.

The product of $235 \times 4$ is 940.

Step 2: The decimal number $2.35$ has 2 digits after the decimal point. So, there are 2 decimal places.

Step 3: In the product 940, count 2 places from the right and place the decimal point. $940. \to 94.0 \to 9.40$.

$9.40$ is the same as $9.4$.

(b) Multiplication of a Decimal by 10, 100, 1000, etc.

Multiplying a decimal number by a power of 10 (like 10, 100, 1000) is a shortcut that involves shifting the decimal point.

Rule: When multiplying a decimal number by 10, 100, 1000, etc., shift the decimal point to the right by a number of places equal to the number of zeros in the power of 10.

If there are not enough digits in the decimal part to shift the decimal point the required number of places, add zeros to the right of the decimal part as needed.

Examples:

• $1.76 \times 10$. 10 has one zero. Shift decimal 1 place right: $17.6$.
• $1.76 \times 100$. 100 has two zeros. Shift decimal 2 places right: $176.$ or $176$.
• $1.76 \times 1000$. 1000 has three zeros. Shift decimal 3 places right. Add a zero: $1.760 \times 1000 \to 1760.$ or $1760$.
• $0.03 \times 100$. 100 has two zeros. Shift decimal 2 places right: $0.03 \to 00.3 \to 003.$ or $3$.
• $5. \times 100 = 5.00 \times 100 = 500$.

(c) Multiplication of a Decimal by another Decimal

To multiply a decimal number by another decimal number:

1. Ignore the decimal points in both numbers and multiply them as if they were whole numbers.
2. Count the total number of decimal places in both of the original decimal numbers combined. This is the sum of the number of digits after the decimal point in the first number and the number of digits after the decimal point in the second number.
3. In the product obtained in Step 1, starting from the rightmost digit, count leftwards by the total number of decimal places calculated in Step 2. Place the decimal point there. If the product has fewer digits than required, add zeros to the left of the product before placing the decimal point.

Example: Multiply $2.5 \times 0.3$.

Step 1: Multiply 25 by 3 as whole numbers: $25 \times 3 = 75$.

Step 2: Count decimal places: $2.5$ has 1 decimal place. $0.3$ has 1 decimal place. Total number of decimal places in the product should be $1 + 1 = 2$.

Step 3: In the product 75, count 2 places from the right and place the decimal point. $75. \to 7.5 \to .75$. Add a leading zero: $0.75$.

Example: Multiply $0.1 \times 0.01$.

Step 1: Multiply 1 by 1 as whole numbers: $1 \times 1 = 1$.

Step 2: Count decimal places: $0.1$ has 1 decimal place. $0.01$ has 2 decimal places. Total number of decimal places = $1 + 2 = 3$.

Step 3: In the product 1, place the decimal point to have 3 decimal places from the right. Add leading zeros: $1. \to .1 \to .01 \to .001$. Add a leading zero before the decimal: $0.001$.

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2. Division of Decimal Numbers

Division of decimal numbers can be performed using long division. The key is to handle the decimal point correctly in the quotient. The process differs slightly depending on whether the divisor is a whole number or a decimal.

(a) Division of a Decimal by a Whole Number

To divide a decimal number (dividend) by a whole number (divisor):

1. Write the long division problem as usual, with the decimal number inside the division symbol and the whole number outside.
2. Perform the division as if the dividend were a whole number.
3. Place the decimal point in the quotient (the result) directly above the decimal point in the dividend. Make sure the digits in the quotient are placed in the correct columns corresponding to the digits in the dividend.
4. If the division does not result in a remainder of 0 after using all the original digits of the dividend, you can add zeros to the right of the last digit in the decimal part of the dividend and continue the division until the remainder becomes 0 or you reach the desired number of decimal places.

Example: Divide $8.4 \div 4$.

Divide 8 by 4 (result 2, write above 8). Place the decimal point in the quotient above the decimal point in 8.4. Bring down 4. Divide 4 by 4 (result 1, write above 4).

Example: Divide $0.35 \div 5$.

Divide 0 by 5 (result 0, write above 0). Place decimal point. Bring down 3. Divide 3 by 5 (result 0, write above 3). Bring down 5. Consider 35. Divide 35 by 5 (result 7, write above 5).

(b) Division of a Decimal by 10, 100, 1000, etc.

Dividing a decimal number by a power of 10 is a shortcut involving shifting the decimal point.

Rule: When dividing a decimal number by 10, 100, 1000, or any higher power of 10, shift the decimal point to the left by a number of places equal to the number of zeros in the divisor (the power of 10).

If there are not enough digits in the whole number part to shift the decimal point, add zeros to the left of the number as needed.

Examples:

• $23.9 \div 10$. 10 has one zero. Shift decimal 1 place left: $2.39$.
• $23.9 \div 100$. 100 has two zeros. Shift decimal 2 places left: $0.239$.
• $23.9 \div 1000$. 1000 has three zeros. Shift decimal 3 places left: $0.0239$.
• $0.7 \div 10$. 10 has one zero. Shift decimal 1 place left: $0.07$.
• $5. \div 100 = 5.0 \div 100$. Shift decimal 2 places left: $0.05$.

(c) Division of a Decimal by another Decimal

To divide a decimal number by another decimal number, the primary step is to transform the problem into an equivalent division where the divisor is a whole number. This is done by shifting the decimal points in both numbers.

Steps:

1. Look at the divisor (the number you are dividing by). Count how many places you need to shift the decimal point to the right to make it a whole number.
2. Shift the decimal point in the dividend (the number being divided) to the right by the exact same number of places you shifted for the divisor. Add zeros to the right of the dividend if necessary to make this shift possible.
3. Now you have a new division problem with a decimal (the new dividend) divided by a whole number (the new divisor).
4. Perform the long division using the method for dividing a decimal by a whole number (Case 2a).
5. Place the decimal point in the quotient directly above the *new* position of the decimal point in the dividend.

Reasoning: This process is based on the fact that multiplying both the numerator and the denominator of a fraction by the same power of 10 does not change its value. For example, $\frac{7.75}{0.25} = \frac{7.75 \times 100}{0.25 \times 100} = \frac{775}{25}$. We convert the division into an equivalent fraction with a whole number denominator.

Example: Divide $7.75 \div 0.25$.

Step 1: Divisor is $0.25$. To make it a whole number (25), shift decimal 2 places to the right.

Step 2: Shift decimal in dividend $7.75$ also 2 places to the right. It becomes $775$.

Step 3: The new division problem is $775 \div 25$.

Perform long division:

Place decimal point above the new position in 775 (at the end). Result is 31.

Example: Divide $0.5 \div 0.05$.

Step 1: Divisor $0.05 \to 5$ (shift 2 places right).

Step 2: Dividend $0.5 \to 0.50 \to 50$ (shift 2 places right, adding a zero).

Step 3: The problem becomes $50 \div 5 = 10$.

Example: Divide $30.94 \div 0.7$.

Step 1: Divisor $0.7 \to 7$ (shift 1 place right).

Step 2: Dividend $30.94 \to 309.4$ (shift 1 place right).

Step 3: The problem becomes $309.4 \div 7$. Perform long division:

Place decimal point in quotient above decimal point in 309.4. Result 44.2.

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Simplification of Expressions Involving Fractions and Decimals

In previous sections, we have learned how to perform basic operations (addition, subtraction, multiplication, and division) with fractions and with decimal numbers separately. Now, we will encounter mathematical expressions that might involve a mix of both fractions and decimals, along with multiple operations and brackets. To simplify such expressions and arrive at the correct single numerical value, we must follow the standard order of operations, usually remembered by the acronym BODMAS.

Furthermore, to simplify expressions involving both fractions and decimals, it is generally a good strategy to convert all the numbers in the expression into a consistent format – either all fractions or all decimals – before performing the operations. The choice of format depends on the specific numbers in the expression and which conversion seems simpler and less likely to introduce repeating decimals that are hard to work with.

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Applying the BODMAS Rule

The BODMAS rule dictates the sequence of operations:

1. B - Brackets: Simplify everything inside brackets first. If there are nested brackets (brackets within brackets), start with the innermost bracket and work outwards. The typical order for nested brackets is Vinculum, then Parentheses $( )$, then Curly Brackets $\{ \}$, and finally Square Brackets $[ ]$.
2. O - Orders: Evaluate powers (like $2^3$) or roots (like $\sqrt{16}$). In the context of Class 7, this might also cover 'Of' used with fractions (e.g., $\frac{1}{2}$ of $20$), which means multiplication.
3. D - Division and M - Multiplication: After simplifying brackets and orders, perform all division and multiplication operations. These two operations have equal priority, so you should perform them from left to right as they appear in the expression.
4. A - Addition and S - Subtraction: Finally, perform all addition and subtraction operations. These two operations also have equal priority and should be performed from left to right as they appear in the expression.

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Strategy for Simplification

When an expression contains both fractions and decimals, follow this general strategy:

1. Choose a Consistent Format:
   • Convert all to Fractions: This is a good approach if the decimals in the expression are terminating and easily convert to fractions with simple denominators (e.g., $0.5 = \frac{1}{2}, 0.25 = \frac{1}{4}, 0.75 = \frac{3}{4}$). Working purely with fractions might avoid issues with non-terminating decimals.
   • Convert all to Decimals: This is a good approach if the fractions in the expression are terminating decimals or if the divisions are easy (e.g., $\frac{1}{2} = 0.5, \frac{3}{4} = 0.75, \frac{1}{5} = 0.2$). Be careful if a fraction results in a non-terminating repeating decimal (like $\frac{1}{3} = 0.333...$); converting such fractions to decimals might lead to approximations unless you are specifically instructed to use decimals and round off.
   Select the format that seems most manageable for the specific numbers in the problem.
2. Rewrite the Expression: Rewrite the entire expression with all numbers in the chosen consistent format.
3. Apply BODMAS: Simplify the expression step-by-step by strictly following the BODMAS rule. Perform operations within brackets, then orders, then division/multiplication (left to right), and finally addition/subtraction (left to right).
4. Simplify Intermediate Results: Simplify fractions at intermediate steps if it makes the calculations easier (e.g., reduce fractions to lowest terms).
5. Final Result: Present the final answer in the simplest form (e.g., lowest terms for a fraction, or a standard decimal format).

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Examples of Simplification

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