Definition 2.1.1. Percent.
A percent is a ratio of part of something to the whole of that thing that is written as parts per hundred.
Section 2.1 Percents
Example 2.1.2.
In a class there are 34 students. Of them 21 are female. The percent is calculated as
\begin{equation*}
\frac{21}{34} = 0.6176\text{.}
\end{equation*}
This number says there are 61 hundreths (remembering our numbering system), so the percent is written as
\begin{equation*}
61.76\%
\end{equation*}
Generally this means we calculate
\begin{equation*}
100 \times \frac{\text{part}}{\text{whole}}.
\end{equation*}
Subsection 2.1.1 Calculating Percents
Example 2.1.3.
In the class there are 34 students. Of them 13 are male. The percent is calculated as
\begin{equation*}
100 \times \frac{13}{34} = 100 \times 0.3824 = 38.24\%.
\end{equation*}
Checkpoint 2.1.4.
In another class there are 78 students and 44 are female. What percent of the students are female?
Answer.
\(56\)
Solution.
A percent is the ratio of part (44 female) to whole (78 total). So this is \(\frac{44}{78} \approx 0.56\text{.}\) This is 56 hundreths, so it is 56%
Note in the first pair of examples we had a whole class of 34 students with 21 female and 13 male. Of course 21+13 = 34, that is the two parts add up to the whole. Because of this 61.76% + 38.24% = 100% as well.
Sometimes we are given the size of the whole and a percent. We must calculate how many in the part.
Example 2.1.5.
In a class of 22 students, 18% are Alaska Native. How many students are Alaska Native?
Solution 1.
We use the same setup as before, but we don’t know the part yet.
\begin{align*}
100 \cdot \frac{P}{22} & = 18\%\\
\frac{P}{22} & = \frac{18}{100}\\
P & = 22 \cdot \frac{18}{100}\\
& \approx 3.96.
\end{align*}
Solution 2.
We know that a percent is a number out of 100, so we can skip a step from the previous example.
\begin{align*}
\frac{P}{22} & = 0.18\\
P & = 22 \cdot 0.18\\
P & = 3.96
\end{align*}
Sometimes we know the size of a part and what percent it is. We can calculate the size of the whole.
Example 2.1.6.
In a class 2 Alaska Native students make up 6.25% of the class. How many students are in the class?
Solution.
Again we use the same setup, but we don’t yet know the whole.
\begin{align*}
\frac{2}{W} & = 0.0625.\\
2 & = 0.0625 \cdot W.\\
\frac{2}{0.0625} & = W.\\
32 & = W.
\end{align*}
Checkpoint 2.1.7.
Find the number of millilitres of alcohol needed to prepare 150 mL of solution that is 5% alcohol.
Answer.
\(7.5\)
Solution.
We need to know what 5% of 150 mL is: \(0.05 \cdot 150 = 7.5\text{.}\)
Checkpoint 2.1.8.
There are 32 students in a class. Below are percents for each racial group tracked. Calculate the number of students in each group.
Solution.
In this class there are \(0.0625 \cdot 32 = 2\) Alaska Native students; \(0.125 \cdot 32 = 4\) Asian students; 2 Black students (same percent as Alaska Native); \(0.7188 \cdot 23 \approx 23.0016\) or 23 White students; and \(0.0938 \cdot 32 \approx 3.0016\) or 3 students who declared as other.
Subsection 2.1.2 Percent Increase/Decrease
A common use of percents is to indicate how much something has increased (or decreased) from one time to the next.
Example 2.1.9.
In spring there were 22 students in a class. In fall there were 34 students in the same class. This was an increase of 34-22=12 students. We can calculate what percent 12 is of 22.
\begin{equation*}
100 \times \frac{12}{22} = 55\%
\end{equation*}
We say that the class size had a percent increase of 55%. Note this says the increase was 55% of the previous whole.
We can think of this in another way.
Example 2.1.10.
In spring there were 22 students in a class. In fall there were 34 students in the same class.
We calculate the percentage the fall class size is of the spring class size.
\begin{equation*}
100 \times \frac{34}{22} = 155\%
\end{equation*}
Because the percent is greater than 100% we know this was an increase. Specifically it was an increase of 55% over the previous semester.
Checkpoint 2.1.11.
What is the percent increase or decrease if enrollment in a class was 57 in spring and 81 in fall?
Answer.
\(42\)
Solution.
The ratio is \(\frac{81}{57} \approx 1.42\text{.}\) Because 81 is greater than 57 this is a percent increase. The increase is 42%.
Checkpoint 2.1.12.
What is the percent increase or decrease if enrollment in a class was 78 in fall and 38 in spring?
Answer.
\(51\)
Solution.
The ratio is \(\frac{38}{78} \approx 0.49\text{.}\) Because 38 is greater than 78 this is a percent decrease. Because the new is 49% of the previous the decrease is 100%-49%=51%.
Subsection 2.1.3 Limitations
When presenting data a percent by itself can be deceptive.
Example 2.1.13.
Which of the following do you suppose represents a greater reduction in students?
| Percent reduction | Total |
| 18% | 495 |
| 1.85% | 54 |
| 60% | 5 |
Solution.
| Percent reduction | Total | Number reduced |
| 18% | 495 | 90 |
| 1.85% | 54 | 1 |
| 60% | 5 | 3 |
The 18% of 495 represents the largest number of students. The 60% is a higher percent, but because the total is so small it represents very few students. A percent is more useful if we also know the total number.
Did you calculate 89 for 18% of 495? Compare the following to see why both are reasonable responses. 89 is what percent of 495? 90 is what percent of 495?
