Topic 5: Order of operations
You might recall a question in the previous section:
Jayne did 5 journeys of 34km and 3 journeys of 57km. She used 31 litres of fuel to cover this distance, how
many km per litre did she obtain. The distance covered was 5 × 34 + 3 × 57 .
There are two possibilities for how this expression is solved. The answer could have been obtained by
working left to right.
5 × 34 + 3 × 57
= 170 + 3 × 57
= 173 × 57
= 9861
If the two multiplication operations were performed first before the addition took place, the answer is
5 × 34 + 3 × 57
= 170 + 171
= 341
To avoid confusion there is a convention that states the order in which operations are performed. When
solving an expression containing different operations use this order.
1. Evaluate brackets or other grouping symbols first. If there are nested brackets (brackets within
brackets) work from the inner set to the outer set.
2. Evaluate any powers or roots. This is covered in the indices module. An example of a simple power
is 32 = 3 × 3 = 9 . An example of a root is
16 = 4 because 4 × 4 = 42 = 16 .
3. Multiplication and division. These operations are equal in priority and an expression containing
both operations should be solved working left to right.
4. Addition and subtraction. These operations are equal in priority and an expression containing both
operations should be solved working left to right.
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For example:
14 − 7 + 2
= 7+2
=9
Addition and subtraction are equal priority, so work from L to R
Subtraction first
then addition
14 − (7 + 2) ÷ 3
= 14 − 9 ÷ 3
= 14 − 3
= 11
Brackets must be simplified first
Division next
then subtraction
The power must be calculated first
The two multiplications next
then addition followed by subtraction
1× 7 + 32 − 3 × 7
=1× 7 + 9 − 3 × 7
= 7 + 9 − 21
= 16 − 21 = − 5
2[ 3 × − 4 − (4 − 10)]
= 2[ 3 × − 4 − − 6]
= 2[ −12 − − 6]
=2[ −12 + 6] =2 × − 6
= −12
Inner most brackets first
Multiplication in the brackets
then addition (remember − − is the same as +)
The operation between the 2 and the bracket is multiply
Modern calculators understand order of operations and will do operations in the correct sequence,
however, you still need to use brackets. The examples above, performed on a calculator are:
14p7+2=9
14p(7+2)P3=11
1O7+3dp3O7=-5
2(3Oz4p(4p10))=-12
Take care when evaluating number expression like this
25 + 42
. The calculator sequence below
4.2 × 60
will produce the incorrect answer;
(25+4d)P4.2O60=585.71…
This is an order of operations problem. To make sure the numerator is divided by the product of
4.2 and 60, the 4.2 x 60 should be bracketed.
(25+4d)P(4.2O60)=0.16269…
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 Video ‘Order of Operations’
Activity
1.
Evaluate the following expressions by pen and paper and calculator methods.
(a)
(d)
20 ÷ (3 + 1)
20 ÷ 2 + (5 − 7)
(b)
(e)
(g)
(4 − 11) × (9 − 7)
(h)
3 × (4 − 6)
3 16 + (6 + 9 ÷ 3)
−
3( 4 − 2 × 8)
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(c)
(f)
(i)
2( 9 − 2 × 3)
14 + (3 + 16 ) × 3
2
−
50 ÷ [ 4 + (2 × 3)]
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