Dividing Fractions — Print
Dividing asks how many of these fit into that, and the answer can be bigger than you started with.
What the practice questions cover
These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Reciprocal — The flip of a fraction. The reciprocal of 35 is 53.
- Divisor — The polynomial you are dividing by.
How many quarter-cup scoops fill three cups?
Twelve. You can count them: four scoops per cup, three cups.
So 3 ÷ 14 = 12. The answer is bigger than the three you started with.
That feels wrong, because dividing is supposed to make things smaller. Here it did not.
Dividing asks how many fit. Small things fit lots of times, so dividing by something under one always gives a bigger answer — and multiplying by the flipped fraction does exactly that.
What dividing really asks
12 ÷ 3 asks how many threes fit into twelve. Dividing by a fraction asks the same question.
3 ÷ 14 asks how many quarters fit into three, and the answer is twelve.
Reading division this way makes the surprising answers stop being surprising.
Small divisor, big answer
Dividing by something under one always gives an answer bigger than what you started with, because small pieces fit in many times.
Flip and multiply
To divide by a fraction, flip it over and multiply instead.
34 ÷ 12 = 34 × 21 = 64 = 32.
Only the second fraction flips. The first one is left exactly as it is.
Only the divisor flips
Flipping both gives a completely different answer. The one being divided by is the one that turns over, and it is always the second.
Why flipping works
Dividing by 14 means asking how many quarters fit — and there are four in every whole.
So dividing by 14 and multiplying by 4 are two ways of asking one question.
The same reasoning works for any fraction, which is why the rule is not a trick.
Check with a picture
Draw the whole, cut it into the pieces named by the divisor, and count. For small numbers the picture is faster than the rule.
Mixed numbers first
A mixed number cannot be flipped as it stands, so turn it into an improper fraction first.
212 becomes 52; its reciprocal is 25.
Convert, divide, then change back to a mixed number at the end if the question asked for one.
Convert before flipping
Flipping 212 to get 12 2 is meaningless. Improper fractions exist exactly so that this step is possible.
Word problems
Ask which number is being cut up and which is doing the cutting. The cutter is the divisor.
"How many 34-metre pieces from 6 metres?" divides 6 by 34, giving 8.
If the answer comes out smaller than expected, the two numbers may be the wrong way round.
Sanity-check the size
Cutting 6 metres into pieces under a metre must give more than six pieces. An answer of 4.5 means the division went the wrong way.
Worked examples
Fraction by fraction
Work out 34 ÷ 12.
- Flip the second fraction: 12 becomes 21. Only the divisor turns over.
- Multiply: 34 × 21 = 64. Tops together, bottoms together.
- Simplify to 32, or 112. How many halves in three quarters? One and a half.
32
A mixed number
Work out 212 ÷ 34.
- Convert: 212 = 52. Mixed numbers cannot be flipped as they stand.
- Flip the divisor and multiply: 52 × 43 = 206. Only the second one flips.
- Simplify to 103 = 313. Bigger than 212, as dividing by a fraction under one must be.
313
A word problem
How many 34-metre ribbons can be cut from 6 metres?
- The ribbon length is doing the cutting, so it is the divisor. 6 ÷ 34.
- Flip and multiply: 6 × 43 = 243. The 6 is 61.
- = 8 ribbons. More than six, since each piece is under a metre.
8
Words to know
- Reciprocal
- The fraction turned upside down. The reciprocal of 34 is 43.
- Dividend
- The number being divided.
- Divisor
- The number you are dividing by — the one that gets flipped.
- Mixed number
- A whole number with a fraction, like 212.
- Improper fraction
- One whose top is bigger than its bottom, like 52.
Mistakes to watch for
Flipping the wrong fraction. In 34 ÷ 12 only the 12 turns over. Flipping the first instead gives 23, which is not even close to the correct 32.
Flipping both fractions. That undoes itself and answers a different question entirely — the rule turns over the divisor alone.
Trying to flip a mixed number directly. 212 must become 52 first; there is no sensible way to invert a whole number sitting beside a fraction.
Expecting the answer to shrink. 3 ÷ 14 = 12, because twelve quarters fit into three. Dividing by anything under one makes the answer bigger.
Dividing the wrong way round in a word problem. Cutting 6 metres into 34-metre pieces is 6 ÷ 34 = 8, not 34 ÷ 6. The piece length is the divisor.
The rule "invert and multiply" has been taught for centuries, and generations have learned it without ever being told why it works.
It works because dividing by 14 and multiplying by 4 ask the same question: how many quarters are in it?
Recipes are the commonest place people meet this outside school — halving a recipe that calls for 23 of a cup needs exactly this arithmetic.
Problem 1
Work out 12 ÷ 14.
Worked answer
2
12 × 41 = 42 = 2
Problem 2
Work out 3 ÷ 14.
Worked answer
12
3 × 4 = 12
Problem 3
Give the reciprocal of 35.
Worked answer
5/3
53
Problem 4
Give the reciprocal of 4.
Worked answer
1/4
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Problem 5
Work out 23 ÷ 2.
Worked answer
1/3
23 × 12 = 26 = 13
Problem 6
Work out 34 ÷ 12.
Worked answer
3/2
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Problem 7
Work out 56 ÷ 23.
Worked answer
5/4
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Problem 8
Work out 212 ÷ 34.
Worked answer
10/3
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Problem 9
How many 34-metre ribbons can be cut from 6 metres?
Worked answer
8
6 × 43 = 8
Problem 10
How many 13-cup scoops fill 4 cups?
Worked answer
12
4 × 3 = 12
Problem 11
Dividing by a fraction smaller than 1 gives an answer that is
- bigger than you started with
- smaller
- the same
- always a whole number
Worked answer
bigger than you started with
3 ÷ 14 = 12
Problem 12
Work out 313 ÷ 112.
Worked answer
20/9
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Problem 13
A 78-litre bottle is shared into 18-litre glasses. How many glasses?
Worked answer
7
78 × 8 = 7
Problem 14
A board 92 feet long is cut into 34-foot pieces. How many pieces?
Worked answer
6
92 × 43 = 366 = 6
Problem 15
Work out 47 ÷ 27.
Worked answer
2
47 × 72 = 2814 = 2
Problem 16
A recipe needs 23 cup of sugar. How many full batches from 4 cups?
Worked answer
6
4 ÷ 23 = 4 × 32 = 6