Fractions: Multiplying and Dividing
Multiplying and dividing fractions uses different rules from adding and subtracting them, and does not require a common denominator: to multiply two fractions, the numerators are multiplied together and the denominators are multiplied together, while to divide by a fraction, the calculation is turned into a multiplication by the reciprocal of the fraction being divided by, a method often remembered as 'keep, flip, change'. Mixed numbers must be converted to improper fractions before either operation is carried out.
Before you start
Make sure you're comfortable with these topics first:
Method
- To multiply two fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.
- Before multiplying, look for a common factor between any numerator and any denominator and cancel it first; this keeps the numbers smaller and means less simplifying is needed at the end.
- To divide by a fraction, 'keep, flip, change': keep the first fraction as it is, flip (find the reciprocal of) the second fraction, and change the divide sign to a multiply sign, then multiply as normal.
- If a question involves a mixed number, convert it to an improper fraction before multiplying or dividing; never multiply or divide the whole-number and fraction parts separately.
- When multiplying or dividing by a whole number, write the whole number as a fraction over 1 first, so the same method applies.
- Once the calculation is complete, simplify the answer fully by dividing the numerator and denominator by their highest common factor, and convert an improper fraction to a mixed number if the question asks for one.
- In a worded problem, decide whether the situation needs multiplication (finding a fraction of a fraction, or scaling) or division (finding how many times one fraction fits into another) before setting up the calculation.
Worked example
Work out 2 and 1/4 divided by 3/5, giving your answer as a mixed number in its simplest form.
- Convert the mixed number to an improper fraction: 2 and 1/4 = 9/4.
- Keep the first fraction, flip the second fraction to its reciprocal, and change divide to multiply: 9/4 divided by 3/5 becomes 9/4 x 5/3.
- Cancel a common factor of 3 between the 9 and the 3: this simplifies to 3/4 x 5/1.
- Multiply the numerators and the denominators: 3 x 5 = 15, and 4 x 1 = 4, giving 15/4.
- Convert the improper fraction to a mixed number: 15/4 = 3 and 3/4.
Practice questions
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Q1Work out 2/5 x 3/7.Show answer
Answer: 6/35
Q2Work out 3/8 x 4/9, giving your answer in its simplest form.Show answer
Answer: 1/6
Q3Work out 5/6 divided by 2/3, giving your answer in its simplest form.Show answer
Answer: 5/4, or 1 and 1/4
Q4Work out 4 x 2/9, giving your answer as a fraction in its simplest form.Show answer
Answer: 8/9
Q5Work out 1 and 3/5 x 2 and 1/2, giving your answer as a mixed number.Show answer
Answer: 4
Q6A water tank is 3/4 full. Ravi uses 2/3 of the water that is in the tank. What fraction of the FULL tank does Ravi use?Show answer
Answer: 1/2 of the full tank
Q7How many 2/5 litre bottles can be filled from 6 litres of juice?Show answer
Answer: 15 bottles
Q8Explain why dividing by 1/2 gives a bigger answer than the number you started with.Show answer
Answer: Dividing by 1/2 is the same as multiplying by its reciprocal, 2, and multiplying a positive number by 2 makes it bigger; dividing by a fraction smaller than 1 always increases the value, because you are asking how many times that smaller amount fits into the original number.
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Work out 3 and 1/3 divided by 5/9, giving your answer as a mixed number in its simplest form.
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A recipe for one batch of muffins needs 2/3 of a cup of sugar. Elena has 6 and 1/2 cups of sugar. (a) Work out how many FULL batches of muffins Elena can make. (b) Work out how much sugar, as a fraction of a cup, Elena has left over after making these full batches.
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Work out 3/4 x 2/9 x 1 and 1/5, giving your answer in its simplest form.
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Free printable worksheet
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This topic is chapter 10 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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