Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical reasoning in HP TET Mathematics. This topic tests your ability to perform arithmetic operations—addition, subtraction, multiplication, and division—on both fractions and decimals, and to convert between the two forms. Questions frequently appear in both the content section and pedagogical contexts, where you may need to identify common student errors.

Mastery here directly supports success in related topics like percentage, ratio-proportion, and mensuration calculations. Expect 3–5 direct questions on this topic, often embedded in word problems involving money, measurement, or data interpretation. The key is speed and accuracy—build fluency through practice rather than memorising procedures mechanically.

Key Concepts

• **Fraction as part of a whole**: A fraction a/b represents 'a' equal parts out of 'b' total parts. The numerator (top) counts parts taken; the denominator (bottom) shows total equal parts.

• **Types of fractions**: Proper fractions have numerator < denominator (3/7). Improper fractions have numerator ≥ denominator (9/4). Mixed fractions combine a whole number with a proper fraction (2¼).

• **Equivalent fractions**: Fractions representing the same value—multiply or divide both numerator and denominator by the same non-zero number (2/3 = 4/6 = 6/9).

• **Decimal place value**: Each digit after the decimal point represents tenths, hundredths, thousandths, etc. In 3.456, the 4 is in tenths place, 5 in hundredths, 6 in thousandths.

• **Terminating vs recurring decimals**: Fractions with denominators having only 2 and 5 as prime factors give terminating decimals (1/4 = 0.25). Others give recurring decimals (1/3 = 0.333...).

• **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions have different denominators and require conversion before addition/subtraction.

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

Formulas / Key Facts

**Conversion formulas**:

• Fraction to decimal: Divide numerator by denominator (3/4 = 3 ÷ 4 = 0.75)
• Decimal to fraction: Write decimal over appropriate power of 10, then simplify (0.125 = 125/1000 = 1/8)
• Mixed to improper: (whole × denominator + numerator)/denominator → 2¾ = (2×4+3)/4 = 11/4
• Improper to mixed: Divide numerator by denominator → quotient is whole part, remainder is new numerator

**Operations on fractions**:

• Addition/Subtraction (unlike): Find LCM of denominators, convert, then add/subtract numerators
• Multiplication: (a/b) × (c/d) = (a×c)/(b×d)
• Division: (a/b) ÷ (c/d) = (a/b) × (d/c) — multiply by reciprocal

**Operations on decimals**:

• Addition/Subtraction: Align decimal points vertically, then proceed as with whole numbers
• Multiplication: Multiply as whole numbers; count total decimal places in both numbers; place decimal point accordingly
• Division: Move decimal point in divisor to make it whole; move same places in dividend; divide normally

**Key facts**:

• Multiplying by 10, 100, 1000 shifts decimal point right by 1, 2, 3 places
• Dividing by 10, 100, 1000 shifts decimal point left by 1, 2, 3 places
• Any fraction can be expressed as a decimal (terminating or recurring)

Worked Examples

**Example 1: Add 2/5 + 3/4**

Step 1: Find LCM of 5 and 4 → LCM = 20

Step 2: Convert to equivalent fractions

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

Step 3: Add numerators → 8/20 + 15/20 = 23/20

Step 4: Convert to mixed fraction → 23/20 = 1 3/20

**Answer: 1 3/20 or 1.15**

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**Example 2: Multiply 0.25 × 1.6**

Step 1: Ignore decimals, multiply as whole numbers → 25 × 16 = 400

Step 2: Count total decimal places → 0.25 has 2 places, 1.6 has 1 place → total 3 places

Step 3: Place decimal point 3 places from right in 400 → 0.400 = 0.4

**Answer: 0.4**

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**Example 3: Divide 3/7 by 9/14**

Step 1: Change division to multiplication by reciprocal

• (3/7) ÷ (9/14) = (3/7) × (14/9)

Step 2: Multiply

• = (3×14)/(7×9) = 42/63

Step 3: Simplify (HCF of 42 and 63 is 21)

• = 42÷21 / 63÷21 = 2/3

**Answer: 2/3**

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**Example 4: Convert 0.875 to a fraction in lowest terms**

Step 1: Write as fraction → 875/1000

Step 2: Find HCF of 875 and 1000 → HCF = 125

Step 3: Divide both by 125 → 875÷125 / 1000÷125 = 7/8

**Answer: 7/8**

Common Mistakes

• **Adding denominators during addition**: Students write 2/5 + 3/4 = 5/9. **Correct approach**: Find common denominator first; never add denominators directly.

• **Forgetting to take reciprocal in division**: Students compute (3/7) ÷ (9/14) as (3×9)/(7×14). **Correct approach**: Always flip the second fraction and multiply.

• **Misplacing decimal point in multiplication**: After multiplying 0.25 × 1.6, students may write 4.0 instead of 0.4. **Correct approach**: Count decimal places in both numbers combined and place point accordingly.

• **Not simplifying final answers**: Leaving 42/63 instead of 2/3. **Correct approach**: Always reduce to lowest terms—examiners may not credit unsimplified fractions.

• **Ignoring decimal alignment in addition**: Adding 3.5 + 0.75 by aligning rightmost digits gives wrong answer. **Correct approach**: Align decimal points vertically (3.50 + 0.75 = 4.25).

• **Converting recurring decimals incorrectly**: Writing 0.333 = 333/1000 instead of 1/3. **Correct approach**: Recognise recurring patterns; 0.3̄ = 1/3 exactly.

Quick Reference

• **Fraction multiplication**: Multiply across—numerator × numerator, denominator × denominator

• **Fraction division**: Flip the second fraction, then multiply

• **Unlike fractions**: LCM of denominators → convert → operate on numerators

• **Decimal × 10ⁿ**: Shift decimal point n places right

• **Decimal to fraction**: Write over power of 10, then simplify using HCF

• **Check your answer**: Convert final fraction to decimal to verify against estimate