Fraction × Fraction
Foundation12 × 13 = 16. Top×top, bottom×bottom.
After this lesson, you can:
- Multiply numerators together and denominators together
- Always simplify the product
- Cancel before multiplying to keep numbers small
1Summary 重點摘要
12 × 13 = 16. Top×top, bottom×bottom.
2Vocabulary cards Vocabulary
P
Product積
The answer after multiplying, e.g. 16 in 12 × 13 = 16.
N
Numerator分子
The top number; numerators multiply together.
D
Denominator分母
The bottom number; denominators multiply together.
S
Simplify before multiplying先約後乘
Cancel diagonally before multiplying, e.g. cancel 2 and 4 in 23 × 14.
O
Of「的」（分數乘法）
"12 of 13" means 12 × 13.
U
Unit fraction單位分數
A fraction with numerator 1, e.g. 16.
C
Cancel約分（對角約）
Divide one numerator and one denominator by a common factor, e.g. 2 and 4 both by 2.
P
Proper fraction真分數
A fraction smaller than 1; multiplying two proper fractions gives an even smaller one.
3Step by step Steps
- Step 1: The ruleTo multiply a fraction by a fraction, multiply the numerators and then multiply the denominators. For 23 × 14, multiply 2 by 1 and 3 by 4, then simplify. Unlike addition, the denominators do not need to match: you can multiply straight away.
- Step 2: Numerator × numeratorStart with the top numbers: 2 × 1 = 2. This product is the numerator of the new fraction, so write it above the line. The top tells how many parts are taken, and the parts multiply just as whole numbers do.
- Step 3: Denominator × denominatorNow the bottom numbers: 3 × 4 = 12. The product is 212. The denominator tells how many equal parts the whole is split into, and after multiplying, the whole is divided into finer parts, so the bottom number grows.
- Step 4: SimplifyThe product 212 is not in simplest form: the top and bottom share the factor 2, so divide both by 2 to get 16. Simplifying keeps the value the same but writes it more simply. Always check whether the answer can be simplified.
- Step 5: Cancel first for easier numbersIn 23 × 14, the numerator 2 and the denominator 4 can both be divided by 2, becoming 1 and 2, so the expression becomes 13 × 12 = 16. The answer is the same, but the numbers stay small and the work is easier.
- Step 6: 'Of' means multiplyA phrase like '12 of 13' is really a multiplication: 12 × 13 = 16. Picture a cake: a third of a half is one sixth of the whole. Whenever you see 'of', write a multiplication sign and calculate.
4Diagrams Diagrams
5Worked example Example
Example: a ribbon is 23 metre long and 14 of it is used. How many metres are used? (Cancel before multiplying)
- Write 23 × 14: the word 'of' means multiply the two fractions.
- Cancel diagonally: numerator 2 and denominator 4 are both divided by 2, becoming 1 and 2.
- The expression becomes 13 × 12: smaller numbers, but the value does not change.
- Multiply the numerators: 1 × 1 = 1, so the product's numerator is 1.
- Multiply the denominators: 3 × 2 = 6, so the product's denominator is 6.
- The product is 16: 1 and 6 share no common factor, so it is already in simplest form.
- Check: without cancelling, 23 × 14 = 212, which simplifies to the same 16.
- Answer: A quarter of two thirds is 23 × 14 = 16, so 16 metre of ribbon is used. Cancelling first keeps the numbers small: 2 and 4 both divide by 2, giving 13 × 12 = 16. Multiplying directly gives 212, which simplifies to the same 16, so the two methods agree and the answer is in simplest form.
6Common mistakes Mistakes
✗ Adding across: 23 × 14 treated as (2+1)/(3+4) = 37.
✓ Multiply: 2 × 1 = 2 and 3 × 4 = 12 → 16.
✗ Leaving 212 unsimplified.
✓ Always finish simplest: 212 = 16.
✗ Cancelling on the same side: in 25 × 23, cancelling the two 2s on top wrongly gives 115.
✓ Cancel diagonally across the two fractions: a numerator with the other fraction's denominator.
✗ Thinking the product is bigger: saying 12 × 13 is bigger than 12.
✓ Multiplying by a proper fraction shrinks: 12 of 13 = 16, which is smaller than 12.
7Quick check Quick check
Q112 × 13 = ?
Answer: 16 (1×1=1, 2×3=6).
Did you get it right?
Q234 × 25 = ? Simplify your answer.
Answer: 620 = 310.
Did you get it right?
Q323 × 35 = ? Simplify your answer.
Answer: 615 = 25 (or cancel the 3s first).
Did you get it right?
Q4What is 14 of 8?
Answer: 2 (14 × 8 = 84 = 2).
Did you get it right?
Q556 × 45 = ? Simplify your answer.
Answer: 2030 = 23.
Did you get it right?
8Recap Recap
- Multiply numerators together and denominators together
- Always simplify the product
- Cancel before multiplying to keep numbers small
- "Of" in fraction problems means multiply
- Multiplying by a proper fraction makes the value smaller