Related Calculations (Edexcel IGCSE Maths)

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Naomi C

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Naomi C

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Related Calculations

What are related calculations?

  • Related calculations allow us to work out answers to difficult problems, using calculations that we already know the answer to
  • If we know a single, simple calculation, we can often use it to find out the answer to many more difficult calculations using related calculations and inverse operations
    • Related calculations use multiples of ten 
    • Inverse operations reverse a calculation that has happened
  • The commutative property can also be used 
    • Adding and multiplying are commutative
      • If a × c, then b × a c and if ac, then ba c
    • Subtracting and dividing are not commutative

What are inverse operations?

  • A mathematical operation is simply the thing that we do to a number to change it to another number
    • Add, subtract, multiply and divide are all examples of operations
  • Inverse operations are simply the thing that we can do to reverse this change
    • Adding and subtracting are inverse operations
    • Multiplying and dividing are inverse operations
  • Inverse operations can be used to find out more tricky calculations quickly from things we already know
    • For example,
      If we know that 3 × 5 = 15, then we also know that 15 ÷ 3 = 5 and 15 ÷ 5 = 3  
      If we know that 32 = 9, then we also know that √9 = 3

How can related calculations be used to simplify problems?

  • If you are given a problem, such as 12 × 13 = 156,  other facts can be quickly deduced
    • 13 × 12 = 156 (commutative law)
      156 ÷ 13 = 12 (inverse operations)
      156 ÷ 12 = 13 (inverse operations)
  • Using multiples of ten can also help to simplify other problems
    • 120 × 13 = (12 × 10) × 13 = 12 × 13 × 10 = 156 × 10 = 1560
    • 1.2 × 13 = (12 ÷ 10) × 13 = 12 × 13 ÷ 10 = 156 ÷ 10 = 15.6
    • 0.013 × 120 = (13 ÷ 1000) × (12 × 10) =13 × 12  ÷ 1000 × 10 = 156 ÷ 100 = 1.56
  • Using a combination of multiples of ten and inverse operations can deduce the answers to many other related calculations
    • 15 600 ÷ 12 = (156 × 100) ÷ 12 = 156 ÷ 12 × 100 = 13 × 100 = 1300
  • If the number you are dividing by is a decimal, use a multiple of ten to change it to an integer before carrying out any calculations
  • Always change both parts of the problem before using related calculations  
    • 1560 ÷ 1.2 = (1560 × 10) ÷ (1.2 × 10)  = 15600 ÷ 12 = 1300
  • This may be easier to see by writing the problem as a fraction
    • fraction numerator 1560 over denominator 1.2 end fraction space equals space 15600 over 12 space equals space fraction numerator 156 space cross times space 100 over denominator 12 end fraction space equals space 13 space cross times space 100 space equals space 1300

Worked example

Given that 43 × 16 = 688, find the answer to

i)
688 ÷ 16

Multiplication is commutative so 43 × 16 = 16 × 43 = 688.

Division is the inverse operation to multiplication so if 16 × 43 = 688 then 688 ÷ 16 = 43.

688 ÷ 16 = 43
ii)
1.6 × 4300

Multiplication is commutative so 43 × 16 = 16 × 43 = 688.

Consider the related calculations.

1.6 = 16 ÷ 10 
4300 = 43 × 100 

Therefore 1.6 × 4300 = (16 ÷ 10) × (43 × 100) = 16 × 43 ÷ 10 × 100.

16 × 43 ÷ 10 × 100 = 688 × 10

1.6 × 4300 = 6880

iii)
68.8 ÷ 4.3

Begin by writing as a fraction and changing the denominator to an integer.

fraction numerator 68.8 over denominator 4.3 end fraction space equals space 688 over 43

Division is the inverse operation to multiplication so if 43 × 16 = 688 then 688 ÷ 43 = 16.

68.8 ÷ 4.3 = 16

iv)
For part (iii) explain how you can use estimation to check your answer
 

Estimate 68.8 ÷ 4.3 by rounding 68.8 to 70 and 4.3 to 5.

70 ÷ 5 = 14

This shows that 16 is likely to be correct, if we had an answer of 1.6 or 160 then we would know we are wrong.

We can estimate 68.8 ÷ 4.3 by carrying out the calculation 70 ÷ 5 = 14 in our heads and comparing our answer

 

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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.