Mixed Numbers and Improper Fractions

Two whole pizzas and three quarters, or eleven quarters. Same pizza either way, and the word improper does more damage here than the maths does.

Fractions · topic 8 of 9Grades 5 to 7Number8 min read7 practice questions

The two forms

Every fraction so far has been part of a single whole. As soon as there is more than one whole in play, there are two perfectly good ways to write the amount down.

  • 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 simply one where the top number is bigger than the bottom. The name is unhelpful, because it suggests something has gone wrong, and children who half-remember that go on to distrust perfectly correct answers of their own for years afterwards.

Seeing both at once

1 whole1 whole34
Two full bars and three quarters of a third bar. Read it as wholes and a part, and it is 2 and three quarters. Count every quarter instead, 4 + 4 + 3, and it is eleven quarters.

Nothing moved between those two readings. The bars are identical, the chocolate is identical, and the only thing that changed was how somebody chose to count it.

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.

That rule is only the picture written out in words. Each whole bar holds 4 quarters and there are 2 whole bars, giving 8 quarters, and the 3 loose ones bring the total to 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.

What you are really asking is how many complete wholes can be built out of 11 quarters. The answer is two, with 3 quarters left over that cannot quite make a third.

Worked examples

1

Mixed to improper

Write 3 and 25 as an improper fraction.
  1. Each whole is 5 fifths, and there are 3 wholes. 3 × 5 = 15.
  2. Add the 2 loose fifths. 15 + 2 = 17.
  3. The pieces are still fifths, so the denominator stays 5.
  4. Sense check: 17 is more than three times 5, so the answer is more than 3 wholes. Correct.
3 and 25 = 175
2

Improper to mixed

Write 236 as a mixed number.
  1. How many sixes fit into 23? 6 × 3 = 18 and 6 × 4 = 24, which is too big.
  2. So there are 3 whole ones, using up 18 of the 23 sixths.
  3. The remainder is 23 − 18 = 5, so 5 sixths are left over.
  4. The denominator stays 6 throughout.
236 = 3 and 56
3

When it comes out exactly

Write 124 as a mixed number.
  1. 12 ÷ 4 = 3 with no remainder.
  2. No remainder means no fraction part is left over at all.
  3. So the answer is simply the whole number 3.
  4. Writing "3 and 04" is not wrong, but nobody writes it that way.
124 = 3

Which form to use

Neither form is better than the other in any general sense. Each one is easier for a particular job, and knowing which job you are doing is the actual skill.

  • 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

2 and 34 becomes 54 because the child adds 2 and 3. The 2 is not two quarters, it is two whole bars, and two whole bars are 8 quarters, so the multiplication has to happen before the addition. Notice that their answer, 54, is barely more than one whole, when they started with nearly three.

2. Changing the denominator

2 and 34 comes out as 118, with the bottom number changed as well as the top. The pieces were quarters before the conversion and they are still quarters afterwards, because nothing in the process cut anything.

3. Treating an improper fraction as a mistake

73 gets crossed out and reworked because the top is bigger than the bottom and that feels like a mistake. It is an entirely ordinary number, worth a little over 2, and the only reason it looks suspicious is the word improper, which was a poor choice of name and has been confusing children ever since.

Practice questions

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For parents

If your child adds the whole number straight to the numerator, repeating the rule will not help, because they were not short of a rule. Draw the bars instead and ask how many quarters are in one whole bar, then in two. The moment they answer eight without pausing, the multiplication in the rule stops being an arbitrary instruction and becomes the only thing that could possibly make sense.

Keep going

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