Improper fractions and mixed numbers: what they are and how to convert

What is an improper fraction?

An improper fraction is a fraction where the numerator is the same as or bigger than the denominator, so it's worth 1 or more. Seven quarters, ⁷⁄₄, is an improper fraction: there are 7 quarters, and 4 quarters make a whole.

A proper fraction is the other kind, with a smaller numerator than denominator, like ³⁄₄. It's always less than 1.

Here's seven quarters drawn as bars. Each bar is one whole, cut into quarters:

What is a mixed number?

A mixed number is a whole number and a proper fraction written together, like 1 ³⁄₄ (one and three quarters). The picture above shows it: one whole bar, and three quarters of the next.

So ⁷⁄₄ = 1 ³⁄₄. Both are right. Which one is better depends on the question: mixed numbers are easier to picture, and improper fractions are often easier to calculate with.

Improper fractions that are whole numbers

When the numerator is a multiple of the denominator, the improper fraction is a whole number. Five fifths is 1, ten fifths is 2, and fifteen fifths is 3. So ¹⁰⁄₅ = 2 and ⁴⁸⁄₆ = 8.

Government guidance for teachers starts here, because it makes the next step easier. If your child can see that 10 fifths is 2 wholes, they can see that 13 fifths is 2 wholes and 3 fifths left over.

How to make an improper fraction into a mixed number

To change an improper fraction into a mixed number, divide the numerator by the denominator. The answer is the number of wholes, and the remainder is how many parts are left over.

Worked example: twenty-one eighths. 8 eighths make a whole. How many wholes are in 21 eighths?

1. Divide: 21 ÷ 8 = 2 remainder 5.
2. That's 2 wholes, with 5 eighths left over.
3. So ²¹⁄₈ = 2 ⁵⁄₈.

Government guidance gives the words for it: "21 eighths is equal to 2 groups of 8 eighths, and 5 more eighths. This is 2 and 5 eighths."

Worked example: seventeen quarters. 17 ÷ 4 = 4 remainder 1, so ¹⁷⁄₄ = 4 ¹⁄₄.

This is division with remainders, which children learn alongside their times tables. The remainder never goes in the whole number part: it becomes the numerator of the fraction.

How do you convert mixed numbers into improper fractions?

To change a mixed number into an improper fraction, multiply the whole number by the denominator, then add the numerator. The denominator stays the same.

Worked example: two and three fifths.

1. Each whole is 5 fifths, so 2 wholes are 2 × 5 = 10 fifths.
2. Add the 3 fifths: 10 + 3 = 13.
3. So 2 ³⁄₅ = ¹³⁄₅.

In the guidance's words: "There are 2 groups of five-fifths, which is 10 fifths and 3 more fifths. This is 13 fifths."

Worked example: four and one eighth. 4 × 8 = 32 eighths, and 32 + 1 = 33, so 4 ¹⁄₈ = ³³⁄₈.

Here a bar model helps. Draw 4 whole bars, each cut into eighths, and one more bar with 1 eighth shaded. Counting the eighths gives 33.

How to convert proper fractions to improper fractions

You can't turn a proper fraction into an improper one without changing its value, because a proper fraction is less than 1 and an improper fraction is 1 or more. When people ask this, they usually mean turning a whole number or a mixed number into an improper fraction.

A whole number works the same way as a mixed number with no fraction part. To write 3 in quarters, multiply: 3 × 4 = 12, so 3 is twelve quarters, ¹²⁄₄. This is useful when adding or subtracting, for example taking a fraction away from a whole number.

Why children need both

In Year 5, answers to fraction sums often go over 1. The national curriculum's own example: ²⁄₅ + ⁴⁄₅ = ⁶⁄₅. Children are then asked to write that as a mixed number: ⁶⁄₅ = 1 ¹⁄₅.

Multiplying fractions by whole numbers does the same. ³⁄₄ × 5 = ¹⁵⁄₄, which is 3 ³⁄₄ as a mixed number. For more on these, see adding and subtracting fractions and multiplying fractions.

In Year 6, children compare and order fractions greater than 1, which is much easier with mixed numbers: see comparing fractions.

Where this fits in the school years

The national curriculum puts converting between mixed numbers and improper fractions in Year 5. Government guidance for teachers suggests starting it in Year 4, straight after placing mixed numbers on a number line, so some schools teach it a year early. Children have already counted past 1 in fractions (one and a quarter, one and a half) since Year 2.

Common mistakes to watch for

• Leaving a whole inside the fraction. Writing ¹⁷⁄₄ as 3 ⁵⁄₄ gives the right amount, but five quarters is more than a whole, so it isn't finished. The answer is 4 ¹⁄₄.
• Changing the denominator. When converting, the denominator stays the same both ways. Two and three fifths is thirteen fifths, not thirteen tenths.
• Adding instead of multiplying. For two and three fifths, some children do 2 + 5 + 3. It's 2 × 5, then add 3.
• Putting the remainder in the wrong place. In 21 ÷ 8 = 2 remainder 5, the 2 is the whole number and the 5 is the numerator.

How to help at home

• Use food. Cut sandwiches into quarters and count them: "We've eaten 6 quarters. How many whole sandwiches is that?"
• Count in fractions past 1. Count up in thirds together: one third, two thirds, one, one and one third, one and two thirds, two.
• Ask it both ways. After "how many wholes in 11 thirds?", ask "how many thirds in 3 and two thirds?"
• Keep the times tables strong. Every conversion is a multiplication or division fact. A printable times table grid helps for checking.

Improper fractions and mixed numbers questions to try

1. Write ¹³⁄₆ as a mixed number.
2. Write ⁴¹⁄₇ as a mixed number.
3. Write 3 ²⁄₃ as an improper fraction.
4. Write 6 ⁴⁄₉ as an improper fraction.
5. A school kitchen has 17 packs of butter, and each weighs a quarter of a kilogram. How much butter is there? Give your answer as a mixed number.
6. A piece of string is 6 ¹⁄₂ m long. How many half-metre lengths can be cut from it?

Answers

1. 2 ¹⁄₆ (13 ÷ 6 = 2 remainder 1)
2. 5 ⁶⁄₇ (41 ÷ 7 = 5 remainder 6)
3. ¹¹⁄₃ (3 × 3 = 9, and 9 + 2 = 11)
4. ⁵⁸⁄₉ (6 × 9 = 54, and 54 + 4 = 58)
5. 4 ¹⁄₄ kg (17 quarters, and 17 ÷ 4 = 4 remainder 1)
6. 13 (6 wholes are 12 halves, and one more half makes 13)