Fractions Notes, Quizzes & Revision

A fraction shows how many equal parts of a whole are taken. It has two parts:

• Numerator (top): how many parts we have.
• Denominator (bottom): how many equal parts the whole is divided into.

Example: 3/4 means 3 parts out of 4 equal parts.

• Proper fraction: numerator < denominator (e.g., 2/5).
• Improper fraction: numerator ≥ denominator (e.g., 7/4).
• Mixed number: whole number + proper fraction (e.g., 1 3/4).
• Equivalent fractions: different fractions equal to the same value (e.g., 1/2 = 2/4 = 3/6).

To find equivalent fractions:

1. Multiply or divide numerator and denominator by the same non-zero number. Example: 2/3 × (2/2) = 4/6.
2. To simplify, divide numerator and denominator by their highest common factor (HCF). Example: 8/12 → HCF = 4 → 8÷4 / 12÷4 = 2/3.

Two common methods:

• Make common denominators: convert fractions to the same denominator and compare numerators. Example: 2/3 and 3/4 → convert to 8/12 and 9/12 → 9/12 is larger → 3/4 > 2/3.
• Cross-multiply: a/b ? c/d → compare ad and bc. If ad > bc then a/b > c/d. Example: 2/3 ? 3/4 → 2×4 = 8 and 3×3 = 9 → 8 < 9 so 2/3 < 3/4.

With same denominators:

Add/subtract numerators; keep the denominator. Example: 3/8 + 2/8 = (3+2)/8 = 5/8.

With different denominators:

1. Find a common denominator (often the LCM).
2. Convert each fraction to an equivalent fraction with that denominator.
3. Add/subtract the numerators and simplify if possible.

Example: 1/6 + 1/4. LCM = 12 → 1/6 = 2/12, 1/4 = 3/12 → 2/12 + 3/12 = 5/12.

• Multiply: multiply numerators and denominators. Example: 2/3 × 3/5 = (2×3)/(3×5) = 6/15 → simplify to 2/5.
• Divide: invert (reciprocal) the divisor and multiply. Example: (2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12 → simplify to 5/6.
• When possible, cancel common factors before multiplying to keep numbers small (cross-cancel).

1. Jane had 3/4 kg of flour. She used 1/3 of it to make mandazi. How much flour did she use?
   1/3 of 3/4 = (1/3) × (3/4) = 3/12 = 1/4 kg.
2. A teacher gives 2/3 of a metre of cloth to one student and 1/4 metre to another. How much cloth did the two students get together?
   2/3 + 1/4 → common denominator 12 → 8/12 + 3/12 = 11/12 metre.

• Always simplify your answer if possible.
• Use models (pie or bar) to understand parts of a whole.
• For addition/subtraction, find the least common denominator for easier calculation.
• When multiplying, simplify (cancel) before multiplying to reduce arithmetic.
• When dividing, remember "invert and multiply".

1. Write three fractions equivalent to 2/5.
2. Simplify: 18/24.
3. Which is larger: 5/8 or 3/5? Show working.
4. Calculate: 3/7 + 2/3.
5. Calculate: (4/9) ÷ (2/3).
6. Word problem: A sack of maize is shared so that A gets 1/2 and B gets 1/3. What fraction remains?