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Mixed numbers and improper fractions

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Some fractions are bigger than one whole. You can write them in two ways: as an improper fraction like 114, or as a mixed number like 2 34. They are the same amount. In this lesson you will learn:

  • the difference between proper fractions, improper fractions and mixed numbers
  • how to see an improper fraction as wholes and parts
  • how to change an improper fraction into a mixed number, and back again
  • where they sit on a number line
  • how to check whether two of them are equal

1. Three kinds of fraction

Key idea Compare the top number with the bottom number. That tells you whether a fraction is less than one whole, or one whole and more.
KindExampleHow to spot itSize
Proper fraction34The top is smaller than the bottom.Less than 1
Improper fraction114The top is the same as or bigger than the bottom.1 or more
Mixed number234A whole number and a proper fraction together.More than 1
Watch out When the top and bottom are the same, the fraction is one whole: 44 = 1 and 99 = 1. These count as improper fractions too.

2. Wholes and parts

Key idea The bottom number tells you how many parts make one whole. Every time you collect that many parts, you have another whole.

Example: 114 is 11 quarters of a pizza

1 whole
1 whole
34
  • 4 quarters make 1 whole pizza, so 8 quarters make 2 whole pizzas.
  • 11 − 8 = 3 quarters are left over.
Answer114 = 234

3. Improper fraction to mixed number

Key idea Divide the top by the bottom. The answer is the whole number. The remainder goes on top of the same bottom number.

Example: change 175 to a mixed number

175=325
  1. How many whole 5s go into 17?17 ÷ 5 = 3 r 2
  2. 3 is the whole number.3
  3. The remainder, 2, goes over the same bottom, 5.325

Two things to look out for

Simplify the fraction part if you can (Lesson 3.1):

144 → 14 ÷ 4 = 3 r 2 → 3 24 = 3 12

No remainder means a whole number:

124 → 12 ÷ 4 = 3 r 0 → 3

4. Mixed number to improper fraction

Key idea Multiply the whole number by the bottom, then add the top. That is the new top. The bottom number stays the same.

Example: change 3 25 to an improper fraction

325=175
  1. Each whole is 5 fifths, so 3 wholes are3 × 5 = 15 fifths
  2. Add the 2 fifths.15 + 2 = 17
  3. Keep the bottom number.175

Whole numbers can be fractions too

A whole number can be written over any bottom number. Multiply to find the top:

3 = 155, because 3 × 5 = 15.

2 = 84 = 63 = 42

Watch out 3 25 is not 325. The 3 and the 2 are not one number: multiply, then add.

5. On a number line

Key idea An improper fraction sits between two whole numbers. The whole number in its mixed number tells you which two: 114 = 2 34, so it is between 2 and 3.

114 on a number line marked in quarters

01234

Count 11 quarters from 0, or go to 2 and then 3 quarters more. Both land on the same place: 114 = 2 34.

6. Equal or not?

Key idea To check whether a mixed number and an improper fraction are equal, change one into the other and compare.

Is 2 13 equal to 73?

  • 2 × 3 = 6, and 6 + 1 = 7. So 2 13 = 73.
AnswerYes, they are equal

The mistakes that cost marks

  1. Joining the digits together. Writing 3 25 as 325. multiply the whole number by the bottom, then add the top: 3 × 5 + 2 = 17, so 175.
  2. Adding instead of multiplying. Writing 3 25 as 55 because 3 + 2 = 5. each whole is 5 fifths, so 3 wholes are 3 × 5 = 15 fifths.
  3. Changing the bottom number. Writing 2 34 as 118. the pieces are still quarters, so the bottom stays 4: 114.
  4. Using the answer to the division as the top. Writing 175 = 3 35. the whole number is the answer (3); the top is the remainder (2): 3 25.
  5. Leaving a remainder written as “r”. Giving 175 as 3 r 2. put the remainder over the bottom number: 3 25.
  6. Not simplifying the fraction part. Giving 3 24 when the question asks for the simplest form. check the fraction part: 2 and 4 are both even, so 3 12.

Quick checklist

Tick each one when you can do it without help.

Download

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Year 4 practice paper PDF · 11+ style, with answer sheet and answers Year 5 practice paper PDF · 11+ style, with answer sheet and answers Year 6 practice paper PDF · 11+ style, with answer sheet and answers

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