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Mathematics — NUMBERS

Subtopic: Fractions (Age: 13, Kenya)

A fraction shows a part of a whole. It has two parts: the numerator (top) and the denominator (bottom).

Fraction form:

3 4 represents "three quarters" or 3/4

1. Types of fractions

Proper fraction: numerator < denominator (example: 2/5).
Improper fraction: numerator ≥ denominator (example: 7/4).
Mixed number: whole number + proper fraction (example: 1 3/4).
Unit fraction: numerator = 1 (example: 1/3).

2. Equivalent fractions

Multiply or divide numerator and denominator by the same non-zero number.

Example: 1/2 = 2/4 = 4/8 because 1×2/2×2 = 2/4 and 1×4/2×4 = 4/8.

3. Simplifying (reducing) fractions

Divide numerator and denominator by their highest common factor (HCF/GCD).

Example: Simplify 12/18. HCF(12,18)=6 → 12÷6 / 18÷6 = 2/3.

4. Comparing fractions

Method 1: Make common denominators. Method 2: Cross-multiply.

Example: Which is bigger, 3/4 or 5/8?
Cross-multiply: 3×8 = 24, 5×4 = 20 → 24 > 20 so 3/4 > 5/8.

5. Addition and subtraction

Like denominators: add/subtract numerators, keep denominator.

2/7 + 3/7 = (2+3)/7 = 5/7

Unlike denominators: find common denominator (usually LCM), convert, then add/subtract.

Example: 1/3 + 1/4. LCM of 3 and 4 is 12 → 1/3 = 4/12, 1/4 = 3/12 → 4/12 + 3/12 = 7/12.

6. Multiplication

Multiply numerators, multiply denominators, then simplify.

Example: (2/5) × (3/4) = (2×3)/(5×4) = 6/20 = 3/10 (after simplifying by 2).

Tip: Cancel common factors between numerators and denominators before multiplying to make calculation easier.

7. Division

To divide by a fraction, multiply by its reciprocal.

Example: (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8 = 1 7/8.

8. Converting between fractions, decimals and percentages

Fraction → decimal: divide numerator by denominator. Example: 3/5 = 3 ÷ 5 = 0.6.
Decimal → percentage: ×100. Example: 0.6 × 100 = 60% → so 3/5 = 60%.
Percentage → fraction: write as /100 and simplify. Example: 25% = 25/100 = 1/4.

9. Mixed numbers and improper fractions

Convert improper fraction to mixed number: divide denominator into numerator.

Example: 11/4 → 11 ÷ 4 = 2 remainder 3 → 2 3/4.
Convert mixed number to improper fraction: (whole × denominator + numerator) / denominator. Example: 2 3/4 = (2×4+3)/4 = 11/4.

10. Worked examples with Kenyan context

Mary buys a chapati for KSh 40. She eats 3/5 of it. How much (KSh) did she eat?
3/5 of 40 = 3/5 × 40 = 3 × 8 = KSh 24.
A plot of land is divided into 8 equal parts. Sam owns 3 parts and Wanjiru owns 1 part. What fraction of the land do they own together?
3/8 + 1/8 = 4/8 = 1/2 (they own half the plot).
Class of 40 learners: 3/5 are girls. How many girls and boys?
Girls = 3/5 × 40 = 24. Boys = 40 − 24 = 16.

11. Practice questions

Simplify 18/24.
Add: 5/6 + 1/4.
Multiply: 4/9 × 3/8.
Divide: 7/10 ÷ 2/5.
Convert 0.75 to a fraction and percentage.
A sugar packet is 1/2 kg. If you use 3/4 of a packet, how much (kg) did you use?

Answers:

18/24 = divide by 6 → 3/4.
5/6 + 1/4 → LCM 12: 10/12 + 3/12 = 13/12 = 1 1/12.
4/9 × 3/8 = 12/72 = 1/6 (after simplifying by 12).
7/10 ÷ 2/5 = 7/10 × 5/2 = 35/20 = 7/4 = 1 3/4.
0.75 = 75/100 = 3/4 = 75%.
1/2 × 3/4 = 3/8 kg.

12. Helpful tips

Always simplify your answer unless asked otherwise.
Use cancellation before multiplying to keep numbers small.
For adding/subtracting, try to use the lowest common multiple of denominators.
Practice with real-life examples like money, ingredients and school items — this helps understanding.

Prepared for Kenyan KCPE/KCSE basic understanding (age 13). Practice more problems to build fluency.

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