1.2 Percents and Proportional Relationships

Introduction

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?[1]

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

Percents

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]

 

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{ percent }\cdot\text{ whole }=\text{ part}[/latex].

To do calculations using percents, we write the percent as a decimal or fraction.

 

Example

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

 

Example

Write each as a percent:

  1. [latex]\displaystyle\frac{1}{4}[/latex]
  2. 0.02
  3. 2.35

 

See the previous two examples worked out in the following video:

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?

Example

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

The following video works through the solutions to the previous two examples:

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.

Example

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

 

To consider a case in which statements that sound contradictory need further consideration, let’s return to the example from the last page.

Example

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?

 

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?

 

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.

Example

A politician’s support increases from 40% of voters to 50% of voters. Describe the change.

 

Second, 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.

 

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

  1. http://www.factcheck.org/cheney_edwards_mangle_facts.html ↵