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Study Guides > Mathematics for the Liberal Arts Corequisite

Percents and Proportional Relationships


What you’ll learn to do: Apply percent and proportional relationships to problems involving rates, money, or geometry

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.  Who was correct? How can we make sense of these numbers?[footnote]http://www.factcheck.org/cheney_edwards_mangle_facts.html[/footnote] Vice President Dick Cheney speaks to service members March 18 at Balad Air Base, Iraq. The vice president came to the base to visit deployed military men and women serving in support of Operation Iraqi Freedom. In this section, we will show how the idea of percent is used to describe parts of a whole.  Percents are prevalent in the media we consume regularly, making it imperative that you understand what they mean and where they come from. We will also show you how to compare different quantities using proportions.  Proportions can help us understand how things change or relate to each other.

Learning Outcomes

  • Given the part and the whole, write a percent
  • Evaluate changes in amounts with percent calculations
  • Calculate both relative and absolute change of a quantity
  • Write a proportion to express a rate or ratio
  • Solve a proportion for an unknown


A percent is a fraction

Recall that a fraction is written [latex]\dfrac{a}{b},[/latex] where [latex]a[/latex] and [latex]b[/latex] are integers and [latex]b \neq 0[/latex]. In a fraction, [latex]a[/latex] is called the numerator and [latex]b[/latex] is called the denominator. A percent can be expressed as a fraction, that is a ratio, of some part of a quantity out of the whole quantity,  [latex]\dfrac{\text{part}}{\text{whole}}[/latex]. Ex. Suppose you take an informal poll of your classmates to find out how many of them like pizza. You find that, out of 25 classmates, 20 of them like pizza. You can represent your findings as a ratio of how many like pizza out of how many classmates you asked.

[latex]\dfrac{20}{25}[/latex] represents the 20 out of 25 classmates who like pizza.

To find out what percent of the 25 asked said they like pizza, divide the numerator by the denominator, then multiply by 100.

[latex]\dfrac{20}{25} = 20 \div 25 = 0.80 = 80 \%[/latex]

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%

convert a percent to a  decimal or fraction

To do mathematical calculations with a given percent, we must first write it in numerical form. A percent may be represented as a percent, a fraction, or a decimal. Convert a percent to a fraction
  1. Write the percent over a denominator of [latex]100[/latex] and drop the percent symbol %.
  2. Reduce the resulting fraction as needed.

Ex. [latex]80 \% =\dfrac{80}{100}=\dfrac{8\cdot 10}{10\cdot 10}=\dfrac{4}{5}[/latex]

Convert a percent to a decimal There are two methods for writing a percent as a decimal.
  1. You can write the percent as a fraction then divide the numerator by the denominator.
  2. Write the percent without the percent symbol %, then place a decimal after the ones place and move it  two places to the left.

Ex. [latex]80 \% =\dfrac{80}{100}=\dfrac{8\cdot 10}{10\cdot 10}=0.8[/latex]

Ex. [latex]80 \% =80.0=0.80=0.8[/latex]



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{ percent }\cdot\text{ whole }=\text{ part}[/latex]. To do calculations using percents, we write the percent as a decimal or fraction.


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.



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


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

  See the previous two examples worked out in the following video: https://youtu.be/Z229RysttR8

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! [ohm_question]17441[/ohm_question]


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.  This example is also worked out in the video below. 


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.  This example is also worked out in the video below.

The following video works through the solutions to the previous two examples: https://youtu.be/QjVeurkg8CQ

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.

Try It

The following example demonstrates how different perspectives of the same information can aid or hinder the understanding of a situation.


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 change 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 change 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.

To consider a case in which statements that sound contradictory need further consideration, let's return to the example from the last 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 the percent of US casualties relative to the coalition casualties (i.e.: coalition casualties used as the base of the percent), while Cheney’s claim was the percent of Iraqi security force and coalition casualties relative to the total number of casualties (i.e.: total casualties used as the base of the percent). Neither statement contradicts the other. It turns out both statistics are in fact fairly accurate.


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="2"][/practice-area]

  Yellow triangle sign of black exclamation mark 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.


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%.

  Second, a caution against averaging percents.


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.  Deciding whether we have enough information to answer the question is a crucial problem solving skill. So we do some additional research and find that the basketball player attempted 200 2-point field goals and 100 3-point field goals. Now, we have enough information to find the player's overall field goal percentage. The player made 200(0.40) = 80 2-point shots and 100(0.30) = 30 3-point shots. Overall, the player made 110 shots out of 300, for a [latex]\displaystyle\frac{110}{300}=0.367=36.7\%[/latex] overall field goal percentage.

  To see these two examples worked out, view the following video: 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.