Mixed Numbers and Improper Fractions
Two whole pizzas and three quarters, or eleven quarters. Same amount of pizza, two ways of writing it, and converting between them takes one line of working.
The two forms
Up to now every fraction has been part of one whole. Once you have more than one whole, there are two ways to write it.
- A mixed number puts the wholes and the leftover part side by side: 2 and 34.
- An improper fraction counts everything in the same size pieces: 114.
An improper fraction is one where the top number is bigger than the bottom. Despite the name, there is nothing wrong with it. It is not an error to be fixed, it is a different way of writing the same number.
Seeing both at once
Nothing changed between those two readings. The bars are the same bars. Only the way of counting changed.
Mixed number to improper fraction
The rule
Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.
For 2 and 34: 2 × 4 = 8, then 8 + 3 = 11, giving 114.
The rule is just the picture in words. Each whole bar is 4 quarters, and there are 2 of them, which is 8 quarters. Then the 3 loose quarters make 11.
Improper fraction to mixed number
The rule
Divide the numerator by the denominator. The whole-number answer is the number of wholes, and the remainder is the new numerator.
For 114: 11 ÷ 4 = 2 remainder 3, giving 2 and 34.
You are asking how many complete wholes you can build from 11 quarters. Two of them, with 3 quarters left over that are not enough for a third whole.
Worked examples
Mixed to improper
- Each whole is 5 fifths, and there are 3 wholes. 3 × 5 = 15.
- Add the 2 loose fifths. 15 + 2 = 17.
- The pieces are still fifths, so the denominator stays 5.
- Sense check: 17 is more than three times 5, so the answer is more than 3 wholes. Correct.
Improper to mixed
- How many sixes fit into 23? 6 × 3 = 18 and 6 × 4 = 24, which is too big.
- So there are 3 whole ones, using up 18 of the 23 sixths.
- The remainder is 23 − 18 = 5, so 5 sixths are left over.
- The denominator stays 6 throughout.
When it comes out exactly
- 12 ÷ 4 = 3 with no remainder.
- No remainder means no fraction part is left over at all.
- So the answer is simply the whole number 3.
- Writing "3 and 04" is not wrong, but nobody writes it that way.
Which form to use
Neither form is better in general. Each one is easier for a different job.
- Mixed numbers are easier to picture and to say. "Two and a half hours" is clearer than "five halves of an hour".
- Improper fractions are easier to calculate with, especially when multiplying and dividing. Convert to improper first, do the work, convert back at the end.
- Read the question. If it says "give your answer as a mixed number", that is a mark for converting.
Common mistakes
1. Adding the whole number to the numerator
Turning 2 and 34 into 54 by doing 2 + 3. The 2 stands for two whole bars, which is 8 quarters, not 2 quarters. Multiply before you add.
2. Changing the denominator
Writing 2 and 34 as 118. The pieces were quarters at the start and they are still quarters at the end. Only the count of them changed.
3. Treating an improper fraction as a mistake
Assuming 73 must be wrong because the top is bigger. It is a completely normal number, worth a bit more than 2. The word "improper" is unfortunate, and it makes children distrust a correct answer.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
Check yourself
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For parents
If your child adds the whole number to the numerator, do not give them the rule again. Draw the bars. Ask how many quarters are in one whole bar, then in two. Once they say eight without hesitating, the rule stops being something to memorise and becomes something obvious.
What to learn next
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