Before this page: know which operation a calculation does first from order of operations, and be able to add and subtract a negative number from adding and subtracting negative numbers.
A calculation need not be worked in the order it is written
4 × 17 × 25 puts the 4 and the 25, which make 100, at opposite ends:
Work out 4 × 17 × 25
The calculation as written
4 × 17 × 25 has two multiplications and no brackets, so it runs from the left.
Only the order changed: 4 × 25 × 17 is the same three numbers, each still multiplied.
Adding and multiplying can be done in any order
8 × 7 × 5 can start from 8 × 7 = 56 or from 7 × 5 = 35, and 56 × 5 and 8 × 35 both come to 280. Adding behaves like this too: any two of the numbers being added can be paired first, and the total is unchanged.
Example: work out 47 + 86 + 53
Pair the two numbers that make a round number.
- 47 + 53 = 100 53 is what 47 needs for a hundred; the 86 is left until then.
- 100 + 86 = 186 Adding 86 to a hundred needs no column addition. Starting with 47 + 86 = 133 does.
The names for the two moves
ContextAn operation is commutative when its two numbers can change places, and associative when either pair of three can be taken first.
Adding and multiplying are both commutative and associative. A GCSE question asks for the reordering itself, not for the name of either property.
Swapping the numbers of a subtraction or a division changes the answer
Subtracting and dividing give a different answer the other way round. 9 − 4 = 5 while 4 − 9 = −5, and 20 ÷ 4 = 5 while 4 ÷ 20 = 0.2. 9 − 4 starts at 9 and takes 4 away; 4 − 9 starts at 4 and takes 9 away, a different question.
Changing which pair is taken first fails for the same two operations. (20 ÷ 4) ÷ 2 = 2.5, but pairing the last two instead gives 20 ÷ (4 ÷ 2) = 10. The ÷ 2 was meant to halve the answer; halving the divisor doubles the answer instead.
Example: work out 72 ÷ 6 ÷ 3
Divide from the left, without pairing the 6 and the 3.
- 72 ÷ 6 = 12 Neither × nor ÷ outranks the other, so with no brackets a line of them is worked from the left.
- 12 ÷ 3 = 4 Pairing 6 ÷ 3 = 2 first would divide 72 by 2 and reach 36, which is nine times too large.
A number moves with the sign in front of it
38 − 17 + 62 can be reordered too:
Work out 38 − 17 + 62
The calculation as written
38 − 17 + 62 starts at 38, takes 17 away and adds 62.
Reordering a calculation
- A number moves with the sign written in front of it.
- Within a run of adding and subtracting, or a run of multiplying and dividing, those numbers may be written in any order.
- The rearranged line is still worked from the left, because reordering adds no brackets.
- The first number in a run carries a + in a sum and a × in a product, even though neither is written.
Example: work out 63 − 28 + 37
Send −28 to the end, so the two numbers being added meet.
- 63 + 37 − 28 63 and 37 meet.
- 63 + 37 = 100 A hundred, in the head.
- 100 − 28 = 72 Taking 28 from a hundred is done in the head; from the left, 63 − 28 = 35 needs an exchange and 35 + 37 = 72 a carry.
A ÷ travels with its number, so cancel before multiplying
A division sign travels with its number in the same way. 36 × 15 ÷ 9 is a run of multiplying and dividing, so the ÷ 9 can be brought forward to meet the 36. Dividing first cancels the 36 down: 36 ÷ 9 = 4, and 4 × 15 = 60.
What moves is the ÷ 9 and not the 9 on its own: 36 × 9 ÷ 15 = 21.6 leaves both signs where they were and answers a different question.
Example: work out 25 × 32 ÷ 8
Cancel before multiplying.
- 32 ÷ 8 = 4 The ÷ 8 moves to the 32, because 8 divides 32 exactly.
- 25 × 4 = 100 Multiplying first reaches 100 as well, but only through 25 × 32 = 800 and then 800 ÷ 8.
Common mistakes
- Turning a subtraction round 24 − 31 starts at 24 and takes 31 away, which carries it past zero to −7; turned round, 31 − 24 = 7.
- Leaving a sign behind when a number moves 54 − 19 + 46 reordered as 54 + 46 + 19 adds the 19 where it should be taken away, and the line comes to 119; with its minus, 54 + 46 − 19 = 81.
- Pairing up divisions In 48 ÷ 4 ÷ 2, 4 ÷ 2 is the easier pair to see, but taking it first divides 48 by 2 instead of dividing the answer by 2: 24 where 48 ÷ 4 ÷ 2 = 6.
Summary
- Adding and multiplying give the same answer whichever order their numbers are taken in.
- A number carries the sign written in front of it wherever it moves, so −17 travels as −17.
- Swapping the two numbers of a subtraction or a division changes the answer: 24 − 31 = −7 and not 7.
- Changing which pair of a division is taken first changes the answer too, so 72 ÷ 6 ÷ 3 is 4 and not 36.
- Choose the order for the arithmetic it makes: pairing 4 and 25 turns 4 × 17 × 25 into a multiplication by 100.