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
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