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Dividing fractions

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Fractions

Dividing fractions

Turn the second fraction upside down and multiply. Strange as it looks, that is genuinely all there is to it.

2/3 ÷ 1/6 asks: how many sixths fit in two-thirds?1/61/61/61/61/61/62/3 of the bar = 4 sixths2/3 × 6/1 = 12/3 = 4 ✓flip the SECOND fraction, then multiply — that flip is the reciprocal
Dividing by less than 1 makes the answer bigger. That is not a mistake.

What dividing by a fraction means

2/3 ÷ 1/6 is asking: 'How many sixths are there in two-thirds?' Draw it and you can count them: the answer is 4. And the rule gives the same thing: 2/3 × 6/1 = 12/3 = 4 ✓

The method

1. Turn the SECOND fraction upside down. 2. Multiply the fractions as usual. 3. Simplify, and write as a mixed number if it is improper. Worked example: 2/5 ÷ 3/10 = 2/5 × 10/3 = 20/15 = 4/3 = 1 1/3

Reciprocals

Turning a fraction upside down gives its RECIPROCAL. The reciprocal of 1/3 is 3/1, which is 3. The reciprocal of 2/5 is 5/2. The reciprocal of 7 is 1/7, because 7 is 7/1. So the rule is really: dividing by a fraction is the same as multiplying by its reciprocal.

Why the answer can get bigger

2/3 ÷ 1/6 = 4 — the answer is much bigger than what you started with. That is not a mistake. Dividing by a number LESS than 1 always makes things bigger, because you are asking how many small pieces fit inside.

Worked examples

2/3 ÷ 1/6

  1. It asks: how many sixths are in two-thirds?
  2. Turn the second fraction upside down: 2/3 × 6/1.
  3. = 12/3 = 4.

Answer: 4

2/5 ÷ 3/10

  1. Turn 3/10 upside down: 2/5 × 10/3.
  2. = 20/15 = 4/3.
  3. Write as a mixed number.

Answer: 1 1/3

The reciprocal of 2/5

  1. Turn it upside down.

Answer: 5/2

The reciprocal of 7

  1. 7 is the same as 7/1.
  2. Turn it over.

Answer: 1/7

Why the answer can get bigger

  1. Dividing by a number less than 1 makes things bigger.
  2. You are asking how many small pieces fit inside.

Answer: Not a mistake — it is what division by a fraction does

1/2 ÷ 1/4

  1. 1/2 × 4/1 = 4/2.

Answer: 2 — there are two quarters in a half

Where you see this in real life

  • How many 1/4-cup scoops fill a 3/4-cup jug? 3/4 ÷ 1/4 = 3 scoops.
  • A 2/3 m ribbon cut into 1/6 m pieces gives 4 pieces — dividing by a smaller fraction gives more of them.

Key words

Reciprocal
a fraction turned upside down
Improper fraction
a fraction whose top is bigger than its bottom
Mixed number
a whole number together with a fraction
Simplify
cancel a fraction down as far as it will go
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