Learning Outcomes
- Given the part and the whole, write a percent
- Calculate both relative and absolute change of a quantity
- Calculate tax on a purchase
In the 2004 vice-presidential debates, Democratic contender John Edwards claimed that US forces have suffered “90% of the coalition casualties” in Iraq. Incumbent Vice President Dick Cheney disputed this, saying that in fact Iraqi security forces and coalition allies “have taken almost 50 percent” of the casualties.[1]
Who was correct? How can we make sense of these numbers?
Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is equivalent to the fraction [latex]\displaystyle\frac{40}{100}[/latex] or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since [latex]\displaystyle\frac{80}{200}=\frac{10}{25}=\frac{40}{100}[/latex].
A visual depiction of 40%
Percent
If we have a part that is some percent of a whole, then [latex]\displaystyle\text{percent}=\frac{\text{part}}{\text{whole}}[/latex], or equivalently, [latex]\text{part}\cdot\text{whole}=\text{percent}[/latex].
To do the calculations, we write the percent as a decimal.
For a refresher on basic percentage rules, using the examples on this page, view the following video.
Examples
In a survey, 243 out of 400 people state that they like dogs. What percent is this?
Example
Write each as a percent:
- [latex]\displaystyle\frac{1}{4}[/latex]
- 0.02
- 2.35
Try It
Throughout this text, you will be given opportunities to answer questions and know immediately whether you answered correctly. To answer the question below, do the calculation on a separate piece of paper and enter your answer in the box. Click on the submit button , and if you are correct, a green box will appear around your answer. If you are incorrect, a red box will appear. You can click on “Try Another Version of This Question” as many times as you like. Practice all you want!
Example
In the news, you hear “tuition is expected to increase by 7% next year.” If tuition this year was $1200 per quarter, what will it be next year?
Try It
Example
The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?
Absolute and Relative Change
Given two quantities,
Absolute change =[latex]\displaystyle|\text{ending quantity}-\text{starting quantity}|[/latex]
Relative change: [latex]\displaystyle\frac{\text{absolute change}}{\text{starting quantity}}[/latex]
- Absolute change has the same units as the original quantity.
- Relative change gives a percent change.
The starting quantity is called the base of the percent change.
For a deeper dive on absolute and relative change, using the examples on this page, view the following video.
The base of a percent is very important. For example, while Nixon was president, it was argued that marijuana was a “gateway” drug, claiming that 80% of marijuana smokers went on to use harder drugs like cocaine. The problem is, this isn’t true. The true claim is that 80% of harder drug users first smoked marijuana. The difference is one of base: 80% of marijuana smokers using hard drugs, vs. 80% of hard drug users having smoked marijuana. These numbers are not equivalent. As it turns out, only one in 2,400 marijuana users actually go on to use harder drugs.[2]
Example
There are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies.
Example
Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?
A video walk-through of this example can be seen here.
Consideration of the base of percentages is explored in this video, using the examples on this page.
Try It
Example
A Seattle Times article on high school graduation rates reported “The number of schools graduating 60 percent or fewer students in four years—sometimes referred to as ‘dropout factories’—decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.”
- Is the “decreased by 17” number a useful comparison?
- Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34?
Example
Let’s return to the example at the top of this page. In the 2004 vice-presidential debates, Democratic candidate John Edwards claimed that US forces have suffered “90% of the coalition casualties” in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies “have taken almost 50 percent” of the casualties. Who is correct?
A detailed explanation of these examples can be viewed here.
Think About It
In the 2012 presidential elections, one candidate argued that “the president’s plan will cut $716 billion from Medicare, leading to fewer services for seniors,” while the other candidate rebuts that “our plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.” Are these claims in conflict, in agreement, or not comparable because they’re talking about different things?
We’ll wrap up our review of percents with a couple cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.
Example
A politician’s support increases from 40% of voters to 50% of voters. Describe the change.
Lastly, a caution against averaging percents.
Example
A basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point of field goal attempts. Find the player’s overall field goal percentage.
For more information about these cautionary tales using percentages, view the following.
