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Section 8.6 Percents

Percents will be used heavily in this module. The percent formula is one that will be used throughout:

\begin{equation*} \frac{part}{whole} = \frac{percent}{100}. \end{equation*}

You can use this formula when you are missing the part, the whole, or the percent.

In 2013-14, there were approximately 49 million students enrolled in public schools. Out of those students, 5.3% were suspended or expelled. How many students were suspended or expelled?

In this case, we know the whole (49 million) and the percent (5.3), but we do not know the part. Therefore,

\begin{equation*} \frac{part}{49} = \frac{5.3}{100} = 0.053, \end{equation*}

so \(part = 0.053 \cdot 49=2.597\text{,}\) so about 2.6 million students were suspended or expelled in 2013-14.

In 2015-16, 43.42% of the students, or 21,700,551 students, were in schools with police officers. How many total students were enrolled in schools in 2015-16?

Now we know the part (21,700,551) and the percent (43.42), but not the whole, so in this case,

\begin{equation*} \frac{21700551}{whole} = 4342/100=0.4342, \end{equation*}

and \(whole = \frac{21700551}{0.4342} = 49978238\text{,}\) so there were 49,978,238 students enrolled in public schools in 2015-16.

In 2015-16, 61,812 students were arrested out of 49,977,268 who were enrolled nationally that year. What percentage of the student population was arrested that year?

We know the part (61,812) and the whole (49,977,268), but not the percent, so

\begin{equation*} \frac{61812}{49977268} = percent/100, \end{equation*}

or \(0.00124 = percent/100\text{,}\) which means that

\begin{equation*} percent = 0.00124 \cdot 100 = 0.124\%, \end{equation*}

so 0.124% of all students were arrested.

Note that the part does not have to be smaller than the whole. For example, 130 is 130% of 100. While in real life we cannot give more than 100% effort, there are situations in which percentages are over 100, for example if comparing two numbers (your pay may be 115% of your friend’s pay) or calculating change (for example, the price of coffee may have gone up 200%).

Percents are often used to represent change. For example, you may read that the number of suspended students in a school increased by 13% in a year. How is this change calculated?

There were 60,170 school arrests in 2013-14 school year, and 61,812 arrests in 2015-16. The total change in the number of school arrests between these two years is 61,812-60,170=1,642. To know what percent increase this is, we need to compare to the original number, in this case 60,170:

\begin{equation*} 1,642/60,170=0.027. \end{equation*}

We convert a decimal to percent by multiplying by 100: since \(part/whole = percent/100\text{,}\) it means that \(percent = part/whole \cdot 100\text{.}\) Therefore, 0.027= 2.72%, so the number of arrests went up by 2.72%.

More generally, the formula for percent change from old value to new value is

\begin{equation*} \frac{end value - start value}{start value}, \end{equation*}

where \(end value - start value\) is total change, and dividing by start value gives percent change. Note that end value does not have to be bigger than start value, in which case we have a decrease and not an increase. For example, if the number of arrests went down from 61,812 to 60,170, then the total change would be -1,642, which is a 1,642/61,812=0.0266=2.66% decrease (since \(0.0266 \cdot 100=2.66\)). Note that in this case we divided by 61,812 because that was the starting value, and the percent decrease is smaller than the percent increase, because the change of is the same in both cases, while the starting value is different.