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 × b = c, then b × a = c and if a + b = c, then b + a = c
- Subtracting and dividing are not commutative
- Adding and multiplying are 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)
- 13 × 12 = 156 (commutative law)
- 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
Examiner Tip
- In a non calculator exam, always use estimation to check your answer is about right
- You do not have to round closely, using 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
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.
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
Begin by writing as a fraction and changing the denominator to an integer.
Division is the inverse operation to multiplication so if 43 × 16 = 688 then 688 ÷ 43 = 16.
68.8 ÷ 4.3 = 16
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