Sequence of Operations - Basic Mathematics for Water and Wastewater Operators - page 90

r ule 5
In a series of divisions, changing the order or the grouping may change the result;
thus, 100 ÷ 10 = 10 but 10 ÷ 100 = 0.1, and (100 ÷ 10) ÷ 2 = 5 but 100 ÷ (10 ÷ 2) = 20.
Again, if no grouping is indicated, the divisions are performed in the order written,
from left to right; thus, 100 ÷ 10 ÷ 2 is understood to mean (100 ÷ 10) ÷ 2.
r ule 6
In a series of mixed mathematical operations, the convention is as follows: Whenever
no grouping is given, multiplications and divisions are to be performed in the order
written, then additions and subtractions in the order written.
SEQUENCE OF OPERATIONS EXAMPLES
In a series of additions, the terms may be placed in any order and grouped in any
way:
4 + 6 = 10 and 6 + 4 = 10
(4 + 5) + (3 + 7) = 19, (3 + 5) + (4 + 7) = 19, and [7 + (5 + 4)] + 3 = 19
In a series of subtractions, changing the order or the grouping of the terms may
change the result:
100 - 20 = 80, but 20 - 100 = -80
(100 - 30) - 20 = 50, but 100 - (30 - 20) = 90
When no grouping is given, the subtractions are performed in the order written, from
left to right:
100 - 30 - 20 - 3 = 47
or by steps:
100 - 30 = 70, 70 - 20 = 50, 50 - 3 = 47
In a series of multiplications, the factors may be placed in any order and in any
grouping:
[(3 × 3) × 5] × 6 = 270 and 5 × [3 × (6 × 3)] = 270
In a series of divisions, changing the order or the grouping may change the result:
100 ÷ 10 = 10, but 10 ÷ 100 = 0.1
(100 ÷ 10) ÷ 2 = 5, but 100 ÷ (10 ÷ 2) = 20