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Study Guides > Mathematics for the Liberal Arts

Percents

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.[footnote]http://www.factcheck.org/cheney_edwards_mangle_facts.html[/footnote] 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].
Rounded rectangle divided into ten vertical sections. The left four are shaded yellow, while the right 6 are empty. 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. https://youtu.be/Z229RysttR8

Examples

In a survey, 243 out of 400 people state that they like dogs. What percent is this?

Answer: [latex]\displaystyle\frac{243}{400}=0.6075=\frac{60.75}{100}[/latex] This is 60.75%. Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.

 

Example

Write each as a percent:
  1. [latex]\displaystyle\frac{1}{4}[/latex]
  2. 0.02
  3. 2.35

Answer:

  1. [latex]\displaystyle\frac{1}{4}=0.25[/latex] = 25%
  2. 0.02 = 2%
  3. 2.35 = 235%

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?

Answer: The tuition next year will be the current tuition plus an additional 7%, so it will be 107% of this year’s tuition: $1200(1.07) = $1284. Alternatively, we could have first calculated 7% of $1200: $1200(0.07) = $84. Notice this is not the expected tuition for next year (we could only wish). Instead, this is the expected increase, so to calculate the expected tuition, we’ll need to add this change to the previous year’s tuition: $1200 + $84 = $1284.

Example

The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?

Answer: To compute the percent change, we first need to find the dollar value change: $6800 – $7400 = –$600. Often we will take the absolute value of this amount, which is called the absolute change: |–600| = 600. Since we are computing the decrease relative to the starting value, we compute this percent out of $7400: [latex]\displaystyle\frac{600}{7400}=0.081=[/latex] 8.1% decrease. This is called a relative change.

 

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. https://youtu.be/QjVeurkg8CQ
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.[footnote]http://tvtropes.org/pmwiki/pmwiki.php/Main/LiesDamnedLiesAndStatistics[/footnote]

Example

There are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies.

Answer: When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is 215 – 75 = 140. From this, we could say “Albertsons has 140 more stores than QFC.” However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base: Using QFC as the base, [latex]\displaystyle\frac{140}{75}=1.867[/latex]. This tells us Albertsons is 186.7% larger than QFC. Using Albertsons as the base,[latex]\displaystyle\frac{140}{215}=0.651[/latex]. This tells us QFC is 65.1% smaller than Albertsons. Notice both of these are showing percent differences. We could also calculate the size of Albertsons relative to QFC:[latex]\displaystyle\frac{215}{75}=2.867[/latex], which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons:[latex]\displaystyle\frac{75}{215}=0.349[/latex], which tells us that QFC is 34.9% of the size of Albertsons.

 

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?

Answer: To answer this question, suppose the value started at $100. After one week, the value dropped by 60%: $100 – $100(0.60) = $100 – $60 = $40. In the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase: $40 + $40(0.75) = $40 + $30 = $70. In the end, the stock is still $30 lower, or [latex]\displaystyle\frac{\$30}{100}[/latex] = 30% lower, valued than it started.

A video walk-through of this example can be seen here. https://youtu.be/4HNxwYMTNl8
Consideration of the base of percentages is explored in this video, using the examples on this page. https://youtu.be/nygw69JqwoQ

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.”
  1. Is the “decreased by 17” number a useful comparison?
  2. Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34?

Answer:

  1. This number is hard to evaluate, since we have no basis for judging whether this is a larger or small change. If the number of “dropout factories” dropped from 20 to 3, that’d be a very significant change, but if the number dropped from 217 to 200, that’d be less of an improvement.
  2. The last sentence provides relative change, which helps put the first sentence in perspective. We can estimate that the number of “dropout factories” was probably previously around 34. However, it’s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate.

 

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?

Answer: Without more information, it is hard for us to judge who is correct, but we can easily conclude that these two percents are talking about different things, so one does not necessarily contradict the other. Edward’s claim was a percent with coalition forces as the base of the percent, while Cheney’s claim was a percent with both coalition and Iraqi security forces as the base of the percent. It turns out both statistics are in fact fairly accurate.

A detailed explanation of these examples can be viewed here. https://youtu.be/Svlu2Lurmsc
 

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?

[practice-area rows="8"][/practice-area]

Yellow triangle sign of black exclamation markWe’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.

Answer: We could describe this using an absolute change: [latex]|50\%-40\%|=10\%[/latex]. Notice that since the original quantities were percents, this change also has the units of percent. In this case, it is best to describe this as an increase of 10 percentage points. In contrast, we could compute the percent change:[latex]\displaystyle\frac{10\%}{40\%}=0.25=25\%[/latex] increase. This is the relative change, and we’d say the politician’s support has increased by 25%.

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.

Answer: It is very tempting to average these values, and claim the overall average is 35%, but this is likely not correct, since most players make many more 2-point attempts than 3-point attempts. We don’t actually have enough information to answer the question. Suppose the player attempted 200 2-point field goals and 100 3-point field goals. Then that player made 200(0.40) = 80 2-point shots and 100(0.30) = 30 3-point shots. Overall, they player made 110 shots out of 300, for a [latex]\displaystyle\frac{110}{300}=0.367=36.7\%[/latex] overall field goal percentage.

For more information about these cautionary tales using percentages, view the following. https://youtu.be/vtgEkQUB5F8

Licenses & Attributions

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  • Problem Solving. Authored by: David Lippman. Located at: http://www.opentextbookstore.com/mathinsociety/. License: CC BY-SA: Attribution-ShareAlike.
  • Caution sign. Authored by: JDDesign. License: CC0: No Rights Reserved.
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  • Review of basic percents. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
  • Absolute and Relative Differences. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
  • Importance of base in percents. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
  • Combining percents. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
  • Evaluating claims involving percents. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
  • Percentage points and averaging percents. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
  • Question ID 17441, 17447, 17443. Authored by: Lippman, David. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.