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GCSEMathsAQARevision Notes1. NumberFractionsOperations with Fractions

Operations with Fractions (AQA GCSE Maths)

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Mark

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Mark

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Adding & Subtracting Fractions

Dealing with mixed numbers

  • Always turn mixed numbers into top heavy fractions before adding or subtracting

Adding & subtracting

  • Adding and subtracting are treated in exactly the same way:
    • Find the lowest common denominator (the smallest whole number that each denominator divides)
    • Write each fraction as an equivalent fraction over this denominator (by multiplying top-and-bottom by the same amount)
    • Add (or subtract) the numerators and write this over a single lowest common denominator
      • do not add the denominators
    • Check for any cancellation (or if asked to turn top heavy fractions back into mixed numbers)

Worked example

(a) Find 2 over 3 plus 1 fifth

Find the lowest common denominator of 3 and 5
 

15 is the smallest number that divides both 3 and 5

the lowest common denominator is 15
 

Write both fractions as equivalent fractions over 15 (by multiplying top and bottom by the same amount)
 

fraction numerator 2 cross times 5 over denominator 3 cross times 5 end fraction plus fraction numerator 1 cross times 3 over denominator 5 cross times 3 end fraction equals 10 over 15 plus 3 over 15
 

Add the numerators and write over a single denominator
 

fraction numerator 10 plus 3 over denominator 15 end fraction equals 13 over 15
 

There is no cancellation

bold 13 over bold 15

(b) Find 3 3 over 4 minus 5 over 8 giving your answer as a mixed number

Change the mixed number into a top heavy fraction (by multiplying the denominator, 4, by the whole number, 3, then adding the numerator, 3)
 

fraction numerator 4 cross times 3 plus 3 over denominator 4 end fraction equals 15 over 4
 

To find 15 over 4 minus 5 over 8 first find the lowest common denominator of 4 and 8
 

8 is the smallest number that divides both 4 and 8

the lowest common denominator is 8
 

Write both fractions as equivalent fractions over 8 (by multiplying top and bottom by the same amount)
 

fraction numerator 15 cross times 2 over denominator 4 cross times 2 end fraction minus fraction numerator 5 cross times 1 over denominator 8 cross times 1 end fraction equals 30 over 8 minus 5 over 8
 

Subtract the numerators and write over a single denominator
 

fraction numerator 30 minus 5 over denominator 8 end fraction equals 25 over 8
 

Change into a mixed number (by dividing 25 by 8 to get 3 remainder 1)
 

equals 3 1 over 8

There is no more cancellation

bold 3 bold 1 over bold 8

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Multiplying Fractions

Dealing with mixed numbers

  • Always turn mixed numbers into top heavy fractions before multiplying

Multiplying fractions

  • Cancel any numbers on the tops of the fractions with numbers on the bottoms of the fractions (either fraction)
  • Multiply the tops
  • Multiply the bottoms
  • Cancel again if possible
  • Turn top heavy fractions back into mixed numbers (if necessary / asked for)

Worked example

Find 4 over 15 cross times 25 over 11 

 

The 15 and 25 can be cancelled before multiplying (to make the next step easier)
 

fraction numerator 4 over denominator 5 cross times 3 end fraction cross times fraction numerator 5 cross times 5 over denominator 11 end fraction equals fraction numerator 4 over denominator up diagonal strike 5 cross times 3 end fraction cross times fraction numerator up diagonal strike 5 cross times 5 over denominator 11 end fraction equals 4 over 3 cross times 5 over 11
 

Multiply the numerators together and the denominators together
 

fraction numerator 4 cross times 5 over denominator 3 cross times 11 end fraction
 

There is no further cancelling that can be done

bold 20 over bold 33

Dividing Fractions

Dealing with mixed numbers

  • Always turn mixed numbers into top heavy fractions before dividing

Dividing fractions

  • Never try to divide fractions
  • Instead “flip’n’times” (flip the second fraction and change ÷ into ×)
  • So divided by fraction numerator space a over denominator b end fraction becomes  cross times fraction numerator space b over denominator a end fraction
  • Then multiply the fractions (multiply tops and multiply bottoms)
  • Cancel the final answer (if possible)

Worked example

Divide 3 1 fourth by 3 over 8, giving your answer as a mixed number

Rewrite 3 1 fourth as an improper fraction

3 1 fourth equals 12 over 4 plus 1 fourth equals 13 over 4

Turn the division into a multiplication, using the fact that dividing by a fraction is the same as multiplying by its reciprocal

13 over 4 divided by 3 over 8 equals 13 over 4 cross times 8 over 3

Multiply the fractions

13 over 4 cross times 8 over 3 equals fraction numerator 13 cross times 8 over denominator 4 cross times 3 end fraction equals 104 over 12

Simplify the fraction, by dividing the numerator and denominator by 4

104 over 12 equals 26 over 3

Rewrite as a mixed number

26 over 3 equals 24 over 3 plus 2 over 3 equals 8 2 over 3

bold 8 bold 2 over bold 3

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    Mark

    Author: Mark

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.