National 5MathsSQARevision NotesNumerical SkillsFractionsMultiplication & Division with Fractions

Multiplication & Division with Fractions (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Multiplying fractions

How do I multiply two fractions?

  • Start by cancelling any factors that are common to both the numerator and the denominator

    • The common factors could be in either of the fractions

    • 3 over 25 cross times 10 over 11 equals 3 over up diagonal strike 25 to the power of 5 cross times up diagonal strike 10 squared over 11 equals 3 over 5 cross times 2 over 11

  • Multiply

    • Multiply the numerators

    • Multiply the denominators

    • 3 over 5 cross times 2 over 11 equals fraction numerator 3 cross times 2 over denominator 5 cross times 11 end fraction equals 6 over 55

  • Cancel any common factors again if possible

How do I multiply two fractions if one is a mixed number?

  • Always convert mixed numbers into improper fractions before multiplying

    • Convert improper fractions back into mixed numbers at the end if required

Worked Example

Evaluate 3 3 over 5 cross times 2 over 9.

Give your answer in simplest form.

Answer:

Change 3 3 over 5 into an improper fraction

  • Multiply the denominator by the whole part and add to the numerator

  • Write this value over the denominator

3 3 over 5 equals fraction numerator 3 cross times 5 plus 3 over denominator 5 end fraction equals 18 over 5

So

3 3 over 5 cross times 2 over 9 equals 18 over 5 cross times 2 over 9

A factor of 9 can be cancelled from the 18 and the 9 before multiplying

table row cell 18 over 5 cross times 2 over 9 end cell equals cell fraction numerator 2 cross times 9 over denominator 5 end fraction cross times fraction numerator 2 over denominator 1 cross times 9 end fraction end cell row blank equals cell fraction numerator 2 cross times up diagonal strike 9 over denominator 5 end fraction cross times fraction numerator 2 over denominator 1 cross times up diagonal strike 9 end fraction end cell row blank equals cell 2 over 5 cross times 2 over 1 end cell end table

Multiply the numerators together and the denominators together

 equals fraction numerator 2 cross times 2 over denominator 5 cross times 1 end fraction equals 4 over 5

There is no further cancelling that can be done, so the answer is in simplest form

4 over 5

Examiner Tips and Tricks

If you didn't cancel the factors of 9 before multiplying in the Worked Example, the answer would proceed

18 over 5 cross times 2 over 9 equals 36 over 45

You would then need to simplify that to 4 over 5 to get full marks.

Dividing fractions

How do I divide two fractions?

  • Flip the second fraction and change ÷ into ×

    • 1 third divided by 4 over 5  becomes  1 third cross times 5 over 4

    • The 'flipped' fraction is called a reciprocal fraction

  • After this proceed as with multiplying fractions

  • Cancel any factors that are common to both the numerator and the denominator

  • Multiply the fractions

    • Multiply the numerators

    • Multiply the denominators

  • Cancel any common factors again if possible

How do I divide two fractions when one of them is a mixed number?

  • Always convert mixed numbers into improper fractions before dividing

    • Convert improper fractions back into mixed numbers at the end if required

Examiner Tips and Tricks

You can remember this process as “Keep Change Flip".

  • Keep the fraction before the division sign the same

  • Change the divide sign to a times

  • Flip the fraction that is after the division sign

Worked Example

Evaluate 2 5 over 6 divided by 7 over 9.

Give your answer in simplest form.

Answer:

Change 2 5 over 6 into an improper fraction

  • Multiply the denominator by the whole part and add to the numerator

  • Write this value over the denominator

2 5 over 6 equals fraction numerator 2 cross times 6 plus 5 over denominator 6 end fraction equals 17 over 6

So

2 5 over 6 divided by 7 over 9 equals 17 over 6 divided by 7 over 9

Turn the division into a multiplication using 'Keep Change Flip'

  • 'Flip' the second fraction and turn ÷ into ×

17 over 6 divided by 7 over 9 equals 17 over 6 cross times 9 over 7

A factor of 3 can be cancelled from the 9 and the 6 before multiplying

table row cell 17 over 6 cross times 9 over 7 end cell equals cell fraction numerator 17 over denominator 2 cross times 3 end fraction cross times fraction numerator 3 cross times 3 over denominator 7 end fraction end cell row blank equals cell fraction numerator 17 over denominator 2 cross times up diagonal strike 3 end fraction cross times fraction numerator 3 cross times up diagonal strike 3 over denominator 7 end fraction end cell row blank equals cell 17 over 2 cross times 3 over 7 end cell end table

Multiply the fractions

equals fraction numerator 17 cross times 3 over denominator 2 cross times 7 end fraction equals 51 over 14

There is no further cancelling that can be done, so the answer is in simplest form

Otherwise, the question does not specify a particular form for the answer, so you can give it as either that improper fraction, or as a mixed number

51 over 14 space space open parentheses or space space 3 9 over 14 close parentheses

Examiner Tips and Tricks

If you didn't cancel the factors of 3 before multiplying in the Worked Example, the answer would proceed

17 over 6 cross times 9 over 7 equals 153 over 42

You would then need to simplify that to 51 over 14 to get full marks.

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.