Year 9 maths lesson · Fractions

Dividing Fractions

Divide by a whole number or fraction. Understand why dividing by something is the same as multiplying by its reciprocal — not just “flip and times”.

  • About 6–9 minutes
  • 9 quick questions as you go
  • Then about 6 minutes of practice
Practice questions → Set for Class

Worked examples

6 ÷ 3 vs 6 × 1/3

  1. 6 ÷ 3 = 2
  2. 6 × 1/3 = 6 thirds = 2
  3. Dividing by 3 is the same as multiplying by 1/3
  4. More generally: dividing by N = multiplying by 1/N

2/3 ÷ 3

  1. Dividing by 3 is timesing by one third
  2. 2/3 × 1/3
  3. Tops: 2 × 1 = 2. Bottoms: 3 × 3 = 9
  4. 2/3 ÷ 3 = 2/9

2 ÷ 1/4 (verify with pizzas)

  1. If dividing by 4 is timesing by 1/4, then dividing by 1/4 must be timesing by 4
  2. 2 × 4 = 8
  3. Check: 4 quarter-slices per pizza × 2 pizzas = 8 quarters
  4. Reciprocal of 1/4 is 4 (flip 1/4 → 4/1 = 4)

5 ÷ 1/3

  1. Multiply by the reciprocal — the reciprocal of 1/3 is 3
  2. 5 × 3 = 15
  3. 5 ÷ 1/3 = 15

4/5 ÷ 2

  1. The reciprocal of 2 is 1/2
  2. 4/5 × 1/2
  3. Tops: 4 × 1 = 4. Bottoms: 5 × 2 = 10
  4. 4/10 simplifies to 2/5

2/3 ÷ 3/4

  1. Multiply by the reciprocal: flip 3/4 to 4/3, change ÷ to ×
  2. 2/3 × 4/3
  3. Tops: 2 × 4 = 8. Bottoms: 3 × 3 = 9
  4. 2/3 ÷ 3/4 = 8/9

2/3 ÷ 4/9

  1. The reciprocal of 4/9 is 9/4
  2. 2/3 × 9/4
  3. Tops: 2 × 9 = 18. Bottoms: 3 × 4 = 12
  4. 18/12 simplifies to 3/2

Before this lesson

It helps to be comfortable with:

Lesson transcript

Everything the lesson says, word for word

Student: Hey Alex, what are we doing today?

Teacher: Dividing fractions. But before we get to fractions, I want to ask you something simple.

Teacher: What’s 6 divided by 3?

Student: Two.

Teacher: And what’s 6 times one third?

Student: That’s… 6 thirds… which is also 2. Wait — they’re the same!

Teacher: Every time. Dividing by 3 is exactly the same as multiplying by one third — they’re not two separate operations, they’re the same thing.

Student: So dividing by any number is just timesing by its fraction version?

Teacher: Yes. Dividing by 3 is times one third. Dividing by 5 is times one fifth. Dividing by N is times one over N.

Teacher: So — 2 thirds divided by 3?

Student: Dividing by 3 is timesing by one third… so 2 thirds times one third. Tops: 2 times 1 is 2. Bottoms: 3 times 3 is 9. Two ninths?

Teacher: Exactly right.

Teacher: Now, what if we divide by a fraction? What’s 2 divided by one quarter?

Student: Hmm… if dividing by 4 is timesing by one quarter… does dividing by one quarter mean timesing by 4?

Teacher: Try it.

Student: 2 times 4 is 8. So… 8?

Teacher: But does that actually make sense? How many quarter slices fit into 2 whole pizzas?

Student: There are 4 quarters in each whole… 2 wholes… that’s 8 quarters. Oh! It IS 8! The rule actually works!

Teacher: There’s a word for what happened to that one quarter. When you flip a fraction upside down, it’s called its reciprocal. The reciprocal of one quarter is 4.

Student: So the reciprocal is just flipping the fraction. And 4 is really four over one.

Teacher: Exactly. Flip the fraction — that’s the reciprocal.

Teacher: So here’s the rule. To divide by a fraction, multiply by its reciprocal. Flip the second fraction and change the divide to a times.

Student: Flip the fraction and change the sign. Got it.

Teacher: Nice. Let’s try a couple. What’s 5 divided by one third?

Student: Multiplying by the reciprocal — the reciprocal of one third is 3. So 5 times 3. That’s 15.

Teacher: Great. Now, 4 fifths divided by 2.

Student: The reciprocal of 2 is one half. So 4 fifths times one half. Tops: 4 times 1 is 4. Bottoms: 5 times 2 is 10. 4 tenths, which simplifies to 2 fifths.

Teacher: Right, try these.

Teacher: Let’s go the whole way. 2 thirds divided by 3 quarters.

Student: Flip 3 quarters — the reciprocal is 4 thirds. So change divide to times…

Student: Tops: 2 times 4 is 8. Bottoms: 3 times 3 is 9. Eight ninths!

Teacher: That’s right. Dividing by something is the same as multiplying by its reciprocal.

Student: Once you’ve done the flip, it’s just multiplying fractions!

Teacher: One more. What’s 2 thirds divided by 4 ninths?

Student: Dividing by 4 ninths is the same as multiplying by its reciprocal — 9 quarters. So 2 thirds times 9 quarters. Tops: 2 times 9 is 18. Bottoms: 3 times 4 is 12. 18 twelfths, which simplifies to 3 halves.

Teacher: Exactly. Right, try these last ones.

More Fractions lessons

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