MULTIPLICATIONS AND DIVISION OF FRACTIONS
Introduction
Fractions are numbers written in the form a/b, where:
- a = numerator (top number)
- b = denominator (bottom number, not zero)
Sometimes we need to multiply or divide fractions in real life — for example: sharing food, measuring ingredients, or finding areas.
Multiplication of Fractions
Definition
Multiplying fractions means finding the product of two or more fractions.
Rule
Multiply the numerators (top numbers).
Multiply the denominators (bottom numbers).
Simplify if possible.
Formula:
(a/b) × (c/d) = (a × c) / (b × d)
Steps
- Multiply the numerators.
- Multiply the denominators.
- Simplify to lowest terms.
- If result is improper (top > bottom), change to a mixed number if needed.
Examples (Multiplication)
Example 1
(2/3) × (4/5) = (2×4)/(3×5) = 8/15
Example 2
(7/8) × (3/4) = (7×3)/(8×4) = 21/32
Example 3 (show simplify)
(5/6) × (2/3) = (5×2)/(6×3) = 10/18 = 5/9 (divide top & bottom by 2)
Example 4 (cancel before multiply)
(4/9) × (9/10) — cancel 9: gives (4/1) × (1/10) = 4/10 = 2/5
Example 5
(1/2) × (3/7) = 3/14
Example 6 (improper result)
(2/5) × (15/4) = (2×15)/(5×4) = 30/20 = 3/2 = 1 1/2
Example 7
(11/12) × (6/11) = cancel 11 → (1/12)×6 = 6/12 = 1/2
Example 8
(5/8) × (8/15) = cancel 8 → 5/15 = 1/3
Example 9
(3/10) × (5/6) = (3×5)/(10×6) = 15/60 = 1/4
Example 10
(9/7) × (2/3) = (9×2)/(7×3) = 18/21 = 6/7
Division of Fractions
Definition
Dividing fractions finds how many times one fraction fits into another.
Rule (Keep – Change – Flip)
Keep the first fraction, change ÷ to ×, flip (take reciprocal of) the second fraction, then multiply.
Formula:
(a/b) ÷ (c/d) = (a/b) × (d/c)
Steps
- Write the first fraction (keep).
- Replace ÷ with × (change).
- Flip the second fraction (reciprocal).
- Multiply numerators and denominators.
- Simplify the result.
Examples (Division)
Example 1
(3/4) ÷ (1/2) = (3/4) × (2/1) = 6/4 = 3/2 = 1 1/2
Example 2
(7/8) ÷ (7/16) = (7/8) × (16/7) = 16/8 = 2
Example 3
(5/6) ÷ (1/3) = (5/6) × (3/1) = 15/6 = 5/2 = 2 1/2
Example 4
(9/10) ÷ (3/5) = (9/10) × (5/3) = 45/30 = 3/2 = 1 1/2
Example 5
(4/7) ÷ (2/3) = (4/7) × (3/2) = 12/14 = 6/7
Example 6
(2/5) ÷ (1/10) = (2/5) × (10/1) = 20/5 = 4
Example 7
(11/12) ÷ (11/6) = (11/12) × (6/11) = 6/12 = 1/2
Example 8
(5/8) ÷ (3/4) = (5/8) × (4/3) = 20/24 = 5/6
Example 9
(3/10) ÷ (9/20) = (3/10) × (20/9) = 60/90 = 2/3
Example 10
(9/7) ÷ (3/14) = (9/7) × (14/3) = 126/21 = 6
Real-Life Uses
- Cooking / Recipes: doubling or halving fractional ingredient amounts.
- Construction: calculating areas or materials measured in fractions of units.
- Sharing: dividing food or money into fractional shares.
- Science: working with ratios and measured quantities.
- Travel / Time: finding fractional parts of journeys or hours.
Tips & Common Mistakes
- Always simplify the final answer (divide numerator & denominator by their GCF).
- For division: remember Keep – Change – Flip (flip the second fraction before multiplying).
- Convert mixed numbers to improper fractions before multiplying or dividing.
- Cancel (cross-cancel) before multiplying to make calculation easier and reduce mistakes.
- Check results for reasonableness (e.g., multiplying two fractions < 1 gives a smaller number; dividing by a small fraction gives a larger result).
Summary
| Operation | Rule |
|---|
| Multiplication | Multiply numerators × numerators, denominators × denominators, then simplify. |
| Division | Keep the first fraction, change ÷ to ×, flip the second (reciprocal), multiply, simplify. |
