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Fractions and Decimals

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Learning objective
Convert between fractions and decimals and perform operations

Introduction

Fractions and decimals are two important ways to represent parts of a whole. Whether you are measuring ingredients for cooking, calculating money in Indian Rupees (INR), or solving problems in exams like the Assam Direct Recruitment Examination (ADRE), understanding fractions and decimals is essential.

In this chapter, you will learn how to convert between fractions and decimals, and how to perform basic arithmetic operations-addition, subtraction, multiplication, and division-on them. These skills will help you solve a variety of practical and exam questions efficiently.

Understanding Fractions and Decimals

What is a Fraction?

A fraction represents a part of a whole. It is written as numerator over denominator, like this: \( \frac{a}{b} \), where:

  • Numerator (top number) shows how many parts we have.
  • Denominator (bottom number) shows into how many equal parts the whole is divided.

For example, if a pizza is cut into 4 equal slices and you eat 1 slice, you have eaten \( \frac{1}{4} \) of the pizza.

What is a Decimal?

A decimal is another way to show parts of a whole, but it uses place values based on powers of ten. For example, the decimal 0.25 means 25 parts out of 100 (since 0.25 = 25/100).

Decimals are written with a decimal point, separating the whole number part from the fractional part. The first digit after the decimal point is the tenths place, the second is the hundredths place, and so on.

Fraction: 3/4 0 0.75 1

Figure: A pie chart divided into 3/4 parts shaded, and a decimal number line showing the equivalent decimal 0.75.

Conversion between Fractions and Decimals

To convert between fractions and decimals, we use division and place value understanding.

graph TD    A[Start with Fraction \frac{a}{b}] --> B[Divide numerator a by denominator b]    B --> C[Result is decimal form]    D[Start with Decimal number] --> E[Count number of digits n after decimal point]    E --> F[Write decimal as \frac{Decimal x 10^n}{10^n}]    F --> G[Simplify the fraction]    C --> H[End]    G --> H

Explanation: To convert a fraction to decimal, divide the numerator by the denominator. To convert a decimal to fraction, count decimal places, write as a fraction with denominator as a power of 10, then simplify.

Addition and Subtraction of Fractions and Decimals

Adding and Subtracting Fractions:

When fractions have the same denominator (called like denominators), simply add or subtract the numerators and keep the denominator the same.

Example: \( \frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7} \)

When denominators are different (unlike denominators), find the Least Common Multiple (LCM) of the denominators to get a common denominator, convert each fraction, then add or subtract.

Adding and Subtracting Decimals:

Align the decimal points vertically and add or subtract digits column-wise, just like whole numbers.

Fractions Decimals
  1. Find LCM of denominators.
  2. Convert fractions to equivalent fractions with LCM as denominator.
  3. Add or subtract numerators.
  4. Simplify if needed.
  1. Write numbers aligning decimal points.
  2. Add or subtract digits column-wise.
  3. Place decimal point in answer aligned with others.
  4. Simplify if needed.

Multiplication and Division of Fractions and Decimals

graph TD    A[Multiplying Fractions] --> B[Multiply numerators]    B --> C[Multiply denominators]    C --> D[Simplify the fraction]    E[Dividing Fractions] --> F[Multiply first fraction by reciprocal of second]    F --> G[Simplify the fraction]    H[Multiplying Decimals] --> I[Multiply as whole numbers]    I --> J[Count total decimal places]    J --> K[Place decimal in product]    L[Dividing Decimals] --> M[Shift decimal points to make divisor whole number]    M --> N[Divide as whole numbers]    N --> O[Place decimal in quotient]

Explanation: For fractions, multiply numerators and denominators directly; for division, multiply by reciprocal. For decimals, multiply as whole numbers and adjust decimal places; for division, shift decimals to make divisor whole number before dividing.

Worked Examples

Example 1: Convert 3/8 to Decimal Easy
Convert the fraction \( \frac{3}{8} \) into decimal form.

Step 1: Divide numerator by denominator: 3 / 8.

Step 2: Perform division: 8 into 3.000 = 0.375.

Answer: \( \frac{3}{8} = 0.375 \).

Example 2: Add 2/3 and 3/4 Medium
Find the sum of \( \frac{2}{3} + \frac{3}{4} \).

Step 1: Find LCM of denominators 3 and 4, which is 12.

Step 2: Convert fractions: \[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}, \quad \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

Step 3: Add numerators: \[ \frac{8}{12} + \frac{9}{12} = \frac{8 + 9}{12} = \frac{17}{12} \]

Step 4: Convert improper fraction to mixed number: \[ \frac{17}{12} = 1 \frac{5}{12} \]

Answer: \( 1 \frac{5}{12} \).

Example 3: Subtract 12.75 - 3.6 Easy
Calculate \( 12.75 - 3.6 \).

Step 1: Align decimal points:
12.75
- 3.60

Step 2: Subtract digits column-wise:
5 - 0 = 5
7 - 6 = 1
2 - 3 (borrow 1 from 1) = 9
1 - 0 = 1

Step 3: Result is 9.15.

Answer: \( 12.75 - 3.6 = 9.15 \).

Example 4: Multiply 5/6 by 2/3 Medium
Calculate \( \frac{5}{6} \times \frac{2}{3} \).

Step 1: Multiply numerators: \( 5 \times 2 = 10 \).

Step 2: Multiply denominators: \( 6 \times 3 = 18 \).

Step 3: Write the fraction: \( \frac{10}{18} \).

Step 4: Simplify by dividing numerator and denominator by 2: \[ \frac{10 \div 2}{18 \div 2} = \frac{5}{9} \]

Answer: \( \frac{5}{9} \).

Example 5: Divide 4.5 by 0.15 Hard
Calculate \( 4.5 \div 0.15 \).

Step 1: Shift decimal points two places right in both numbers to make divisor whole number: \[ 4.5 \rightarrow 450, \quad 0.15 \rightarrow 15 \]

Step 2: Divide 450 by 15: \[ 450 \div 15 = 30 \]

Answer: \( 4.5 \div 0.15 = 30 \).

Formula Bank

Fraction to Decimal Conversion
\[ \text{Decimal} = \frac{\text{Numerator}}{\text{Denominator}} \]
where: Numerator = top number of fraction, Denominator = bottom number of fraction
Decimal to Fraction Conversion
\[ \text{Fraction} = \frac{\text{Decimal} \times 10^n}{10^n} \]
where: Decimal = given decimal number, n = number of digits after decimal point
Addition of Fractions
\[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \]
where: a,b = numerator and denominator of first fraction; c,d = numerator and denominator of second fraction
Subtraction of Fractions
\[ \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \]
where: a,b = numerator and denominator of first fraction; c,d = numerator and denominator of second fraction
Multiplication of Fractions
\[ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \]
where: a,b = numerator and denominator of first fraction; c,d = numerator and denominator of second fraction
Division of Fractions
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \]
where: a,b = numerator and denominator of first fraction; c,d = numerator and denominator of second fraction

Tips & Tricks

Tip: When adding or subtracting fractions, always find the Least Common Multiple (LCM) of denominators first.

When to use: Adding or subtracting fractions with unlike denominators.

Tip: To convert decimals to fractions quickly, count decimal places and use powers of 10 as denominator.

When to use: Converting terminating decimals to fractions.

Tip: For multiplying fractions, simplify crosswise before multiplying to reduce calculation.

When to use: Multiplying fractions to make calculations easier.

Tip: When dividing decimals, multiply numerator and denominator by \( 10^n \) to remove decimal points.

When to use: Dividing decimals to convert into whole number division.

Tip: Align decimal points vertically when adding or subtracting decimals to avoid errors.

When to use: Adding or subtracting decimal numbers.

Common Mistakes to Avoid

❌ Adding fractions by simply adding numerators and denominators directly.
✓ Find common denominator first, then add adjusted numerators.
Why: Students confuse fraction addition with whole number addition.
❌ Ignoring decimal point alignment when adding or subtracting decimals.
✓ Always align decimal points vertically before performing operations.
Why: Misalignment leads to incorrect place value addition or subtraction.
❌ Not simplifying fractions after operations.
✓ Always reduce fractions to simplest form for clarity and correctness.
Why: Students overlook simplification, leading to non-standard answers.
❌ Dividing decimals without shifting decimal points in divisor to make it whole number.
✓ Multiply both dividend and divisor by \( 10^n \) to eliminate decimals in divisor.
Why: Students perform incorrect division leading to wrong answers.
❌ Confusing multiplication and division rules for fractions.
✓ Remember to multiply numerators and denominators for multiplication; multiply by reciprocal for division.
Why: Misunderstanding operation rules causes calculation errors.
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