Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical literacy at the primary and upper-primary levels. For MP TET, this topic tests both your conceptual clarity and your ability to perform quick, error-free calculations. Questions typically involve simplification, conversion between fractions and decimals, and word problems requiring operations on mixed numbers.

Mastery here is essential because fractions and decimals reappear in percentage, ratio-proportion, and mensuration problems. The pedagogy section may also ask how children develop fraction sense, so understanding the "why" behind procedures matters as much as the "how." Expect 3–5 direct questions plus indirect application in other arithmetic topics.

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Key Concepts

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

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

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

• **Lowest terms (simplest form)**: A fraction is in lowest terms when the HCF of numerator and denominator is 1.

• **Decimal place value**: Each place to the right of the decimal point represents tenths, hundredths, thousandths, and so on—each place is 1/10 of the previous.

• **Terminating vs non-terminating decimals**: A fraction in lowest terms gives a terminating decimal only if the denominator has no prime factors other than 2 and 5.

• **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions do not. Converting to like fractions is necessary before adding or subtracting.

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Formulas / Key Facts

| Operation | Rule | |-----------|------| | **Addition/Subtraction of fractions** | Convert to like fractions (common denominator), then add/subtract numerators. | | **Multiplication of fractions** | (a/b) × (c/d) = (a × c) / (b × d). Simplify by cancelling common factors first. | | **Division of fractions** | (a/b) ÷ (c/d) = (a/b) × (d/c). Multiply by the reciprocal of the divisor. | | **Fraction → Decimal** | Divide numerator by denominator. Example: 3/4 = 3 ÷ 4 = 0.75. | | **Decimal → Fraction** | Write decimal over the appropriate power of 10 and simplify. Example: 0.625 = 625/1000 = 5/8. | | **Mixed → Improper** | Whole × Denominator + Numerator, over the same denominator. Example: 2 3/5 = (2×5 + 3)/5 = 13/5. | | **Improper → Mixed** | Divide numerator by denominator; quotient is whole part, remainder is new numerator. |

**Quick conversion benchmarks** (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

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Worked Examples

### Example 1: Addition of Unlike Fractions

**Problem**: Add 2/3 and 5/6.

**Solution**: 1. Find LCM of denominators: LCM(3, 6) = 6. 2. Convert: 2/3 = 4/6 (multiply numerator and denominator by 2). 3. Add numerators: 4/6 + 5/6 = 9/6. 4. Simplify: 9/6 = 3/2 = 1 1/2.

**Answer**: 1 1/2 or 1.5

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### Example 2: Multiplication Involving Mixed Numbers

**Problem**: Multiply 1 2/5 by 2 1/3.

**Solution**: 1. Convert to improper fractions: 1 2/5 = 7/5; 2 1/3 = 7/3. 2. Multiply: (7/5) × (7/3) = 49/15. 3. Convert to mixed number: 49 ÷ 15 = 3 remainder 4 → 3 4/15.

**Answer**: 3 4/15

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### Example 3: Decimal to Fraction Conversion

**Problem**: Express 0.875 as a fraction in lowest terms.

**Solution**: 1. Write as fraction: 0.875 = 875/1000. 2. Find HCF(875, 1000) = 125. 3. Divide both: 875 ÷ 125 = 7; 1000 ÷ 125 = 8.

**Answer**: 7/8

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### Example 4: Word Problem

**Problem**: A ribbon is 3.6 m long. If 1 1/4 m is cut off, what length remains?

**Solution**: 1. Convert 1 1/4 to decimal: 1.25 m. 2. Subtract: 3.6 − 1.25 = 2.35 m.

**Answer**: 2.35 m (or 2 7/20 m)

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Common Mistakes

1. **Adding denominators when adding fractions**

• Wrong: 2/5 + 1/3 = 3/8.
• Correct: Find common denominator first → 6/15 + 5/15 = 11/15.

2. **Forgetting to invert the divisor in division**

• Wrong: (3/4) ÷ (2/5) = (3/4) × (2/5).
• Correct: Multiply by reciprocal → (3/4) × (5/2) = 15/8.

3. **Misplacing the decimal point in multiplication**

• Wrong: 0.3 × 0.2 = 0.6.
• Correct: Count total decimal places (1 + 1 = 2) → 0.06.

4. **Not simplifying the final answer**

• Exam answers are expected in lowest terms. Always check for common factors.

5. **Confusing mixed-number subtraction when borrowing is needed**

• When subtracting 3 1/4 − 1 3/4, borrow 1 from the whole: 3 1/4 = 2 5/4 → 2 5/4 − 1 3/4 = 1 2/4 = 1 1/2.

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Quick Reference

• **Like fractions**: Same denominator → directly add/subtract numerators.
• **Multiplication shortcut**: Cancel common factors across numerators and denominators before multiplying.
• **Division rule**: Keep-Change-Flip (keep first fraction, change ÷ to ×, flip second fraction).
• **Decimal places in multiplication**: Total decimal places in product = sum of decimal places in factors.
• **Decimal places in division**: Shift decimal in divisor to make it a whole number; shift equally in dividend.
• **Benchmark fractions**: Memorise 1/2, 1/4, 1/5, 1/8 equivalents to speed up mental conversion.