Mixed numbers and improper fractions: practice and how to convert

What the two forms mean

In a proper fraction, like 3/4, the numerator is smaller than the denominator, so the amount is less than 1. In an improper fraction, like 7/4 or 4/4, the numerator is equal to or bigger than the denominator, so the amount is 1 or more. A mixed number, like 1 3/4, writes the same amount as a whole number and a proper fraction side by side.

Neither form is more correct than the other. 7/4 and 1 3/4 are the same number, the way 60 minutes and 1 hour are the same length of time.

Picture it with fraction bars

Draw three bars, each cut into 4 equal parts. Shade all of the first two bars and 3 parts of the third. That shaded amount is 2 3/4: two wholes and three quarters.

Now count every shaded quarter. Each whole bar has 4, so the two full bars give 2 × 4 = 8, and the third bar adds 3 more. That makes 11 quarters, or 11/4. This is the whole method in one picture: multiply the wholes by the denominator, then add the extra parts.

A number line shows the same thing. Start at 0 and make 11 jumps of 1/4. You pass 1 after 4 jumps and 2 after 8 jumps, and land three jumps past 2, at 2 3/4.

Converting mixed numbers to improper fractions

Multiply the whole number by the denominator, add the numerator, and write the result over the same denominator. The denominator never changes, because the size of the pieces doesn't change.

• 1 1/2: 1 × 2 + 1 = 3, so 1 1/2 = 3/2.
• 4 1/3: 4 × 3 + 1 = 13, so 4 1/3 = 13/3.
• 2 5/6: 2 × 6 + 5 = 17, so 2 5/6 = 17/6.
• 5 5/8: 5 × 8 + 5 = 45, so 5 5/8 = 45/8.

Converting improper fractions to mixed numbers

Divide the numerator by the denominator. The number of times it goes in is the whole number, and whatever is left over becomes the new numerator. This is just asking how many full wholes you can make from the pieces.

• 23/6: 6 goes into 23 three times (18) with 5 left over, so 23/6 = 3 5/6.
• 9/2: 2 goes into 9 four times (8) with 1 left over, so 9/2 = 4 1/2.
• 16/4: 4 goes into 16 exactly four times, so 16/4 = 4, a whole number.
• 14/8: 8 goes into 14 once with 6 left over, so 14/8 = 1 6/8, which simplifies to 1 3/4.

When to use each form

Improper fractions are easier to calculate with, especially when multiplying or dividing. To work out 2 1/2 × 3, change 2 1/2 to 5/2 first: 5/2 × 3 = 15/2. Mixed numbers are easier to picture, so they suit final answers and measurements. Most people know at a glance what 7 1/2 inches looks like, but 15/2 inches takes a moment.

How it's taught in school

In the US, the Common Core brings in mixed numbers in grade 4 (standard 4.NF.B.3), where children break a mixed number into wholes and parts, such as 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8, and add and subtract mixed numbers with like denominators. In grade 5 they meet mixed numbers again in word problems and when multiplying.

In England, the national curriculum asks Year 5 pupils to recognize mixed numbers and improper fractions and convert from one form to the other. Year 5 pupils also multiply proper fractions and mixed numbers by whole numbers, so the two skills grow together.

How parents can help

• Use a measuring cup. A recipe needs 2 1/2 cups of flour and you only have a 1/2 cup scoop. How many scoops? Five, so 2 1/2 = 5/2.
• Count pizza slices. If every pizza is cut into 8 slices and 13 slices are eaten, that is 13/8, or 1 5/8 pizzas.
• Ask for both forms. Whenever your child gives one, ask for the other.
• Check by converting back. Changing the answer back to the starting form catches most slips.

The practice set below gives fresh questions in both directions, checked as you go, with hints and worked steps. To practice on paper, print a fresh worksheet with an answer key.