Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical reasoning at the primary level and appear consistently in Assam TET Paper I Mathematics. This topic tests a candidate's ability to perform operations—addition, subtraction, multiplication and division—on fractions and decimals, and to convert between the two forms. Mastery here is essential because these concepts reappear in percentage, ratio-proportion and measurement problems.

For the Assam TET, expect direct calculation questions as well as word problems set in everyday contexts (sharing food, measuring cloth, money transactions). You must be fluent with like and unlike fractions, mixed numbers, and place-value reasoning for decimals. Errors in this topic often stem from careless handling of denominators or misplaced decimal points—both are heavily tested.

Key Concepts

• **Fraction as part of a whole**: A fraction a/b means a equal parts out of b total parts. The numerator (a) counts parts; the denominator (b) names the size of each part.

• **Types of fractions**: Proper (numerator < denominator), improper (numerator ≥ denominator) and mixed numbers (whole + proper fraction). Convert between them freely.

• **Equivalent fractions**: Multiplying or dividing numerator and denominator by the same non-zero number gives an equivalent fraction. Example: 2/3 = 4/6 = 6/9.

• **Like vs unlike fractions**: Like fractions share the same denominator; unlike fractions do not. To add or subtract unlike fractions, first find a common denominator (usually the LCM).

• **Decimal place value**: Each place to the right of the decimal point represents tenths, hundredths, thousandths and so on. The value decreases by a factor of 10 at each step.

• **Fraction-decimal link**: A fraction can be converted to a decimal by dividing numerator by denominator. A terminating decimal can be written as a fraction with denominator 10, 100, 1000 etc.

• **Comparing fractions and decimals**: Convert to a common form (same denominator or same number of decimal places) before comparing.

Formulas / Key Facts

| Operation | Fractions | Decimals | |-----------|-----------|----------| | Addition/Subtraction | Convert to like fractions, then add/subtract numerators | Align decimal points, then add/subtract column by column | | Multiplication | (a/b) × (c/d) = (a × c) / (b × d) | Multiply ignoring decimals, then place decimal point so total decimal places = sum of decimal places in factors | | Division | (a/b) ÷ (c/d) = (a/b) × (d/c) | Make divisor a whole number by shifting decimal in both dividend and divisor equally, then divide |

**Conversion facts**

• Fraction → Decimal: Divide numerator by denominator.
• Decimal → Fraction: Write decimal over appropriate power of 10 and simplify. Example: 0.75 = 75/100 = 3/4.
• Mixed number → Improper: (whole × denominator) + numerator, over denominator.
• Improper → Mixed: Divide numerator by denominator; quotient is whole part, remainder over denominator is fraction part.

**Common decimal-fraction equivalents to memorise**

• 1/2 = 0.5
• 1/4 = 0.25, 3/4 = 0.75
• 1/5 = 0.2, 2/5 = 0.4, 3/5 = 0.6, 4/5 = 0.8
• 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875

Worked Examples

**Example 1 — Adding unlike fractions**

Add 2/5 and 3/4.

Step 1: Find LCM of 5 and 4 → 20.

Step 2: Convert each fraction.

• 2/5 = (2 × 4)/(5 × 4) = 8/20
• 3/4 = (3 × 5)/(4 × 5) = 15/20

Step 3: Add numerators → 8 + 15 = 23.

Answer: 23/20 or 1 3/20.

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**Example 2 — Multiplying decimals**

Multiply 2.4 × 0.35.

Step 1: Ignore decimals and multiply 24 × 35 = 840.

Step 2: Count total decimal places in factors → 1 (in 2.4) + 2 (in 0.35) = 3.

Step 3: Place decimal point 3 places from the right in 840 → 0.840 = 0.84.

Answer: 0.84.

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**Example 3 — Dividing a fraction by a fraction**

Divide 5/6 by 2/3.

Step 1: Take reciprocal of divisor → 3/2.

Step 2: Multiply → (5/6) × (3/2) = 15/12.

Step 3: Simplify → 15/12 = 5/4 = 1 1/4.

Answer: 5/4 or 1.25.

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**Example 4 — Decimal division**

Divide 4.56 by 0.8.

Step 1: Shift decimal one place right in both numbers to make divisor whole → 45.6 ÷ 8.

Step 2: Perform division → 45.6 ÷ 8 = 5.7.

Answer: 5.7.

Common Mistakes

1. **Adding fractions by adding numerators and denominators separately** Wrong: 1/2 + 1/3 = 2/5. Fix: Find common denominator first → 3/6 + 2/6 = 5/6.

2. **Forgetting to simplify the final answer** Wrong: Leaving 4/8 as final answer. Fix: Always reduce to lowest terms → 4/8 = 1/2.

3. **Misaligning decimal points during addition/subtraction** Wrong: Adding 3.5 + 0.25 as 3.5 + 0.25 = 0.60 (treating 3.5 as 0.35). Fix: Write 3.50 + 0.25, align points → 3.75.

4. **Placing decimal incorrectly after multiplication** Wrong: 0.6 × 0.3 = 1.8 (wrong decimal placement). Fix: 6 × 3 = 18; two decimal places total → 0.18.

5. **Inverting the wrong fraction in division** Wrong: (a/b) ÷ (c/d) taken as (b/a) × (c/d). Fix: Invert only the divisor → (a/b) × (d/c).

Quick Reference

• To add/subtract fractions: same denominator first, then operate on numerators.
• Multiplying fractions: multiply across (numerator × numerator, denominator × denominator).
• Dividing fractions: multiply by the reciprocal of the divisor.
• Decimal multiplication: total decimal places in product = sum of decimal places in factors.
• Decimal division: shift decimals equally to make divisor a whole number.
• Always simplify fractions and check decimal point placement before marking your answer.