Multiply & Divide Fractions

Elementary Math
Skill: fractions-multiply-divide

Multiply & Divide Fractions

Multiply fractions straight across (cancel first to keep numbers small), and divide with keep-change-flip — then simplify every answer.

1 Multiply straight across — divide with keep-change-flip

Multiplication is direct; division turns into multiplication with one flip.

Multiply: straight across

23 × 34 = 2×33×4 = 612 = 12

Multiply the tops, multiply the bottoms, then simplify.

Cancel FIRST to keep numbers small

In 23 × 34, the 3s cancel before multiplying: 24 = 12. Any top can cancel with any bottom — that is the same as simplifying at the end, just easier.

Divide: keep-change-flip

23 ÷ 49 = 23 × 94 = 1812 = 112

Keep the first fraction, change ÷ to ×, flip the second (its reciprocal). Then multiply.

Why keep-change-flip works

Dividing by 14 asks “how many quarters fit?” Four quarters fit in 1, so 1 ÷ 14 = 4 — exactly what 1 × 41 gives. Flipping counts how many of the divisor fit.

2 See it: 2/3 × 1/2 as an area

Shade 2/3 of the width, then 1/2 of the height. The overlap is the product.

2/3 of the width shaded1/2 of the height shaded → overlap = 2/6 = 1/3
The grid has 3 × 2 = 6 cells; 2 overlap: 2/3 × 1/2 = 2/6 = 1/3.

3 Worked examples

Straight across, cancel first, keep-change-flip.

Example 1 Multiply: 2/3 × 3/4
  1. Thinking: cancel first: the 3s cancel (one top, one bottom).
  2. Now multiply: 2 × 1 = 2 on top; 1 × 4 = 4 on the bottom.
  3. 2/4 simplifies to 1/2.
2/3 × 3/4 = 1/2
Example 2 Divide: 1/2 ÷ 1/4
  1. Thinking: keep-change-flip: 1/2 × 4/1.
  2. Multiply: 1 × 4 = 4; 2 × 1 = 2 → 4/2 = 2.
  3. Sense check: two quarters fit in one half. Correct.
1/2 ÷ 1/4 = 2
Example 3 Of a number: 3/5 of 10
  1. Thinking: “of” means multiply: 3/5 × 10.
  2. Write 10 as 10/1: (3 × 10)/(5 × 1) = 30/5 = 6.
  3. Cancel first if you like: 10/5 = 2, then 3 × 2 = 6.
3/5 of 10 = 6
Example 4 Divide with simplifying: 2/3 ÷ 4/9
  1. Thinking: keep-change-flip: 2/3 × 9/4.
  2. Cancel first: 2 with 4 (→ 1 and 2), 3 with 9 (→ 1 and 3).
  3. Multiply: (1 × 3)/(1 × 2) = 3/2 = 1 1/2.
2/3 ÷ 4/9 = 1 1/2

4 Common mistakes

Two traps catch almost everyone.

Mistake 1: cross-multiplying when you should multiply straight across
Wrong
23 × 34 = 89 — cross-multiplied by habit.
Right
Multiply straight across: 612 = 1/2. Cross-multiplication is for comparing, not multiplying.
Fix: multiplication goes straight across; cross products belong to comparisons.
Mistake 2: flipping the wrong fraction
Wrong
23 ÷ 49 = 32 × 49 — flipped the first fraction.
Right
Keep the first fraction; flip the second: 23 × 94 = 112.
Fix: chant it: keep (first), change (the sign), flip (the second).

5 Key vocabulary

Reciprocal
A fraction flipped: the reciprocal of 3/4 is 4/3.
Keep-change-flip
Keep the first fraction, change ÷ to ×, flip the second.
Cancel
Dividing a top and a bottom by a common factor before multiplying.
Of
In fraction problems, “of” means multiply: 1/2 of 10 = 5.

6 Quick check

Try these, then reveal the answers.

1. What is 1/2 × 2/5?
Answer
1/5 (cancel the 2s: 1/5).
2. What is the reciprocal of 3/4?
Answer
4/3 (flip it).
3. What is 3/4 ÷ 1/2?
Answer
1 1/2 (3/4 × 2/1 = 6/4 = 3/2).

Key points to remember

  • Multiply straight across: tops × tops, bottoms × bottoms.
  • Cancel first: any top with any bottom, before multiplying.
  • Divide with keep-change-flip (flip the second fraction only).
  • Simplify every answer; write improper results as mixed numbers.
  • “Of” means multiply: 3/5 of 10 = 6.
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