Fractions and Decimals

Overview

Fractions and decimals form the backbone of primary mathematics and appear consistently in OTET Paper I. This topic tests your conceptual understanding of part-whole relationships and your ability to perform arithmetic operations accurately. Questions typically involve addition, subtraction, multiplication, and division of fractions and decimals, along with conversion between the two forms.

For aspiring primary teachers, mastery of this topic is essential not just for clearing the exam but for effective classroom teaching. Students often struggle with fractions and decimals because these concepts require a shift from whole-number thinking. Understanding common misconceptions helps you both answer pedagogy-linked questions and teach more effectively.

Expect 3-5 direct questions from this topic, often combined with word problems involving money, measurement, or ratio-proportion contexts.

Key Concepts

• **Fraction as part of a whole**: A fraction a/b represents 'a' equal parts out of 'b' total parts. The numerator tells how many parts we have; the denominator tells how many equal parts make the whole.

• **Types of fractions**: Proper fractions (numerator < denominator), improper fractions (numerator ≥ denominator), and mixed numbers (whole number + proper fraction). Example: 3/4 is proper, 7/4 is improper, 1¾ is mixed.

• **Equivalent fractions**: Fractions that represent the same value. Multiply or divide both numerator and denominator by the same non-zero number. Example: 2/3 = 4/6 = 6/9.

• **Like and unlike fractions**: Like fractions have the same denominator; unlike fractions have different denominators. Converting to like fractions is essential before adding or subtracting.

• **Decimal as fraction with denominator 10, 100, 1000...**: 0.7 = 7/10, 0.35 = 35/100. Place value determines the denominator.

• **Place value in decimals**: Tenths (first place after decimal), hundredths (second place), thousandths (third place). Example: In 3.257, the 2 is in tenths place, 5 in hundredths, 7 in thousandths.

• **Relationship between fractions and decimals**: Every fraction can be converted to a decimal by dividing numerator by denominator. Terminating decimals have denominators with only 2 and 5 as prime factors.

Formulas / Key Facts

**Fraction Operations:**

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

**Decimal Operations:**

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

**Conversions:**

• Fraction to decimal: Divide numerator by denominator
• Decimal to fraction: Write decimal over appropriate power of 10, simplify
• Mixed to improper: (whole × denominator + numerator)/denominator
• Improper to mixed: Divide numerator by denominator; quotient is whole part, remainder is new numerator

Worked Examples

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

Step 1: Find LCM of denominators 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 = 1 3/20

**Example 2: Multiply 0.25 × 1.6**

Step 1: Multiply as whole numbers. 25 × 16 = 400 Step 2: Count decimal places. 0.25 has 2 places, 1.6 has 1 place. Total = 3 places. Step 3: Place decimal point. 400 → 0.400 = 0.4

**Example 3: Divide 3/4 by 2/5**

Step 1: Write the reciprocal of divisor. Reciprocal of 2/5 is 5/2. Step 2: Multiply. 3/4 × 5/2 = 15/8 Step 3: Convert to mixed number. 15/8 = 1 7/8

**Example 4: Convert 0.375 to a fraction in lowest terms**

Step 1: Write as fraction. 0.375 = 375/1000 Step 2: Find HCF of 375 and 1000. HCF = 125. Step 3: Simplify. 375÷125 / 1000÷125 = 3/8

Common Mistakes

• **Adding numerators and denominators separately** → Wrong: 1/2 + 1/3 ≠ 2/5. Correct: Find common denominator first. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

• **Forgetting to align decimal points in addition/subtraction** → Students write 2.5 + 0.75 as 2.5 + 75 and get wrong answers. Correct: Write as 2.50 + 0.75 = 3.25.

• **Multiplying denominators when adding fractions** → Students confuse addition with multiplication rules. Remember: Only multiply denominators when multiplying fractions, not when adding.

• **Misplacing decimal point after multiplication** → Count decimal places in both original numbers, not just one. In 0.3 × 0.2, answer has 2 decimal places: 0.06, not 0.6.

• **Inverting the wrong fraction in division** → When dividing a/b ÷ c/d, invert only the divisor (c/d becomes d/c). Never invert the dividend.

• **Not simplifying final answers** → Always reduce fractions to lowest terms. 4/8 should be written as 1/2.

Quick Reference

• To add/subtract unlike fractions: LCM → equivalent fractions → operate on numerators
• To multiply fractions: Multiply across (numerator × numerator, denominator × denominator)
• To divide fractions: Keep-Change-Flip (keep first, change ÷ to ×, flip second)
• Decimal places in multiplication = sum of decimal places in both factors
• 1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 3/4 = 0.75, 1/8 = 0.125 (memorise these)
• Mixed number to improper: Multiply whole by denominator, add numerator, keep same denominator