The four operations
The order of operations
Work out brackets first, then powers (indices), then multiplication and division, then addition and subtraction.
Multiplication and division have equal rank, so work left to right between them; the same for addition and subtraction.
- 7 + 3 × 4
- multiplication first: 3 × 4 = 12
- 7 + 12 = 19 (not 10 × 4 = 40)
Negative numbers
- Adding a negative is subtracting: 5 + (−3) = 2.
- Subtracting a negative is adding: −7 − (−3) = −7 + 3 = −4.
- Multiplying or dividing: same signs give a positive, different signs give a negative. (−6) × (−4) = 24, (−20) ÷ 5 = −4.
Long multiplication and using a fact you know
Split the second number into tens and ones, multiply by each, then add.
- 347 × 26
- 347 × 6 = 2082
- 347 × 20 = 6940 (the 2 is worth 20)
- 2082 + 6940 = 9022
If you know 23 × 47 = 1081, then 2.3 × 4.7 = 10.81: each number was divided by 10, so the answer is divided by 100.
The same fact works for division:
- from 23 × 47 = 1081 you know 1081 ÷ 47 = 23
- Dividing by 4.7 instead of 47 means dividing by a number ten times smaller, so the answer is ten times bigger
- 1081 ÷ 4.7 = 230
Multiplying decimals
Ignore the decimal points, multiply the whole numbers, then put the point back.
The answer has as many decimal places as the two numbers have between them.
- 4.3 × 0.26
- ignore the points: 43 × 26 = 1118
- decimal places: 1 in 4.3 and 2 in 0.26, so 3 in the answer
- 4.3 × 0.26 = 1.118
Long division by chunking
To divide by a two-digit number, take away big chunks of it (10 lots, 20 lots, ...) until what is left is small, then count the lots.
- 1564 ÷ 34
- 34 × 40 = 1360, and 1564 − 1360 = 204
- 34 × 6 = 204, and 204 − 204 = 0
- 40 + 6 = 46, so 1564 ÷ 34 = 46
- check: 34 × 46 = 1564
Working backwards and problems in steps
"I think of a number, multiply it by 4, then add 9.
The answer is 37." To find the number, undo each step with its inverse (opposite) operation, in reverse order: the inverse of adding is subtracting, and the inverse of multiplying is dividing.
- the answer was 37
- undo + 9: 37 − 9 = 28
- undo × 4: 28 ÷ 4 = 7
- the number was 7 (check: 7 × 4 + 9 = 37)
A money problem is several small calculations in a row. Write each one down.
- Lee buys 3 cakes at £1.85 each and 2 drinks at £0.95 each, and pays with a £20 note
- cakes: 3 × £1.85 = £5.55
- drinks: 2 × £0.95 = £1.90
- total: £5.55 + £1.90 = £7.45
- change: £20 − £7.45 = £12.55
Counting combinations
With 4 starters and 6 mains there are 4 × 6 = 24 different meals, because each starter goes with each main.
This is the product rule: multiply the numbers of choices.
With a third choice, such as 5 desserts, multiply again: 4 × 6 × 5 = 120 meals.
Check the order of operations
Type 7 + 3 × 4 into the calculator exactly as written, then (7 + 3) × 4. Then try 20 − 8 ÷ 4 and (20 − 8) ÷ 4. Before each one, say what you expect. A scientific calculator follows the order of operations; where you need something else, you add brackets.