Related Calculations

What are related calculations?

Related calculations allow us to work out answers to difficult problems using more simple calculations
If we know a single, simple calculation, we can often use it to find out the answer to many more difficult calculations
- Related calculations use multiples of ten
- Inverse operations reverse a calculation that has happened
The commutative property (where the order of numbers in a calculation does not matter), can also be used
- Adding and multiplying are commutative
  - 3 × 2 = 6 and 2 × 3 = 6
  - 3 + 2 = 5 and 2 + 3 = 5
- Subtracting and dividing are not commutative
  - 8 - 2 = 6 but 2 - 8 = -6
  - 8 ÷ 2 = 4 but 2 ÷ 8 = 0.25

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

An inverse operation is the thing that we can do to reverse this change

Adding is the inverse of subtracting (and vice versa)
Multiplying is the inverse of dividing (and vice versa)

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
  - $\frac{1560}{1.2} = \frac{15600}{12} = \frac{156 \times 100}{12} = 13 \times 100 = 1300$

Exam Tip

In a non calculator exam, always use estimation to check your answer is about right.
- Rounding numbers to the nearest ten will still let you know whether your answer has the correct number of zeros or whether the decimal place is in the correct place.

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

$\frac{68.8}{4.3} = \frac{688}{43}$

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

Related Calculations (Edexcel GCSE Maths: Foundation)

Revision Note