Basic Math Operations - Mathematics and Statistics for the Environment - page 42

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

If no grouping is indicated, the divisions are performed in the order written—from left to right:

100 ÷ 5 ÷ 2 is understood to mean (100 ÷ 5) ÷ 2

In a series of mixed mathematical operations, the rule of thumb is that, whenever no grouping is
given, multiplications and divisions are to be performed in the order written, then additions and
subtractions in the order written.
Consider the following classic example of sequence of operations (Stapel, 2012):

Problem: Simplify 4 - 3[4 - 2(6 - 3)] ÷ 2.

Solution:

4 - 3 [4 - 2(6 - 3)] ÷ 2

4 - 3[4 - 2(3)] ÷ 2

4 - 3[4 - 6] ÷ 2

4 - 3[-2] ÷ 2

4 + 6 ÷ 2

4 + 3 = 7

2.4 PERCENT
The word “percent” means “by the hundred.” Percentage is usually designated by the symbol %;
thus, 15% means 15 percent or 15/100 or 0.15. These equivalents may be written in the reverse order:
0.15 = 15/100 = 15%. In wastewater treatment, percent is frequently used to express plant perfor-
mance and for control of biosolids treatment processes. When working with percent, the following
key points are important:

• Percents are another way of expressing a part of a whole.
• Percent means “by the hundred,” so a percentage is the number out of 100. To determine
percent, divide the quantity we wish to express as a percent by the total quantity, then
multiply by 100:

Part
Whole

Percent (%) =

(2.1)

For example, 22 percent (or 22%) means 22 out of 100, or 22/100. Dividing 22 by 100
results in the decimal 0.22:

22
100

22 % == 022 .