Dividing fractions: practice problems with answers

What dividing by a fraction means

Division asks how many of one thing fit into another. 12 ÷ 3 asks how many threes fit into 12. In the same way, 3 ÷ 1/2 asks how many halves fit into 3.

Picture three bars, each cut into two halves. Count the halves: there are 6. So 3 ÷ 1/2 = 6. That is why dividing by a fraction less than 1 gives an answer bigger than the number you started with: small pieces fit in many times.

Why keep, change, flip works

Start with unit fractions. Every whole holds 4 quarters, so asking how many quarters fit into 5 is the same as working out 5 × 4 = 20. Dividing by 1/4 is multiplying by 4, which is 4/1, the flipped fraction.

Now try 4 ÷ 2/3. First count thirds: 4 wholes hold 4 × 3 = 12 thirds. Each group of 2/3 uses 2 of those thirds, so there are 12 ÷ 2 = 6 groups. You multiplied by 3 and divided by 2, which is exactly multiplying by 3/2. Keep, change, flip is a shortcut for those two moves.

You can always check a division by multiplying back. In the example above, 3/4 ÷ 2/5 = 15/8, and 15/8 × 2/5 = 30/40 = 3/4. The Common Core uses the same idea: 2/3 ÷ 3/4 = 8/9 because 3/4 of 8/9 is 2/3.

Dividing fractions with whole numbers

There are two directions, and both use the same steps once the whole number is written over 1.

• Fraction ÷ whole number: 1/3 ÷ 2 = 1/3 × 1/2 = 1/6. Sharing a third of a pizza between 2 people gives each a sixth.
• Fraction ÷ whole number, another way: 4/5 ÷ 2 = 4/5 × 1/2 = 4/10 = 2/5. Halving four fifths gives two fifths.
• Whole number ÷ unit fraction: 5 ÷ 1/4 = 5 × 4 = 20. Five cups of rice fill 20 quarter-cup scoops.
• Whole number ÷ any fraction: 6 ÷ 3/4 = 6/1 × 4/3 = 24/3 = 8. Six cups of flour make 8 batches if each batch needs 3/4 cup.

Dividing fractions practice problems, worked through

• 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. Two quarters fit into a half.
• 5/6 ÷ 1/3 = 5/6 × 3/1 = 15/6 = 5/2, which is 2 1/2.
• 2/3 ÷ 4/9 = 2/3 × 9/4 = 18/12 = 3/2, which is 1 1/2.
• 2 1/4 ÷ 3/8: change 2 1/4 to 9/4, then 9/4 × 8/3 = 72/12 = 6. Check: 6 × 3/8 = 18/8 = 9/4.

Cover the answers and try them first. Then use the practice set below for fresh questions, each checked straight away, with hints and worked steps if you get stuck.

How it's taught in school

In the US, the Common Core starts in grade 5 with dividing a unit fraction by a whole number, like 1/3 ÷ 4, and a whole number by a unit fraction, like 4 ÷ 1/5 (standard 5.NF.B.7). In grade 6, students divide fractions by fractions and solve word problems with them (6.NS.A.1), using fraction models and story contexts before the shortcut.

In England, Year 6 pupils divide proper fractions by whole numbers. Dividing by a fraction comes in Key Stage 3, from Year 7, where pupils use all four operations with proper and improper fractions.

How parents can help

• Use measuring cups. How many 1/3 cup scoops make 2 cups? Fill and count: 6, so 2 ÷ 1/3 = 6.
• Say it in words. Reading 6 ÷ 3/4 as "how many three quarters fit into 6?" makes the answer feel sensible.
• Estimate first. Dividing by a number less than 1 should give a bigger answer.
• Check by multiplying back. If the answer times the divisor gives the starting number, it is right.

For practice on paper, print a fresh worksheet with an answer key, with answers only or answers with worked steps so your child can see where a slip happened.