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Mathematics LibreTexts

5.2.1: Solving Percent Problems

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  • Page ID 62169

  • The NROC Project

Learning Objectives

  • Identify the amount, the base, and the percent in a percent problem.
  • Find the unknown in a percent problem.

Introduction

Percents are a ratio of a number and 100, so they are easier to compare than fractions, as they always have the same denominator, 100. A store may have a 10% off sale. The amount saved is always the same portion or fraction of the price, but a higher price means more money is taken off. Interest rates on a saving account work in the same way. The more money you put in your account, the more money you get in interest. It’s helpful to understand how these percents are calculated.

Parts of a Percent Problem

Jeff has a coupon at the Guitar Store for 15% off any purchase of $100 or more. He wants to buy a used guitar that has a price tag of $220 on it. Jeff wonders how much money the coupon will take off the original $220 price.

Problems involving percents have any three quantities to work with: the percent , the amount , and the base .

  • The percent has the percent symbol (%) or the word “percent.” In the problem above, 15% is the percent off the purchase price.
  • The base is the whole amount. In the problem above, the whole price of the guitar is $220, which is the base.
  • The amount is the number that relates to the percent. It is always part of the whole. In the problem above, the amount is unknown. Since the percent is the percent off , the amount will be the amount off of the price.

You will return to this problem a bit later. The following examples show how to identify the three parts: the percent, the base, and the amount.

Identify the percent, amount, and base in this problem.

30 is 20% of what number?

Percent: The percent is the number with the % symbol: 20%.

Base : The base is the whole amount, which in this case is unknown.

Amount: The amount based on the percent is 30.

Percent=20%

Base=unknown

The previous problem states that 30 is a portion of another number. That means 30 is the amount. Note that this problem could be rewritten: 20% of what number is 30?

Identify the percent, base, and amount in this problem:

What percent of 30 is 3?

The percent is unknown, because the problem states " What percent?" The base is the whole in the situation, so the base is 30. The amount is the portion of the whole, which is 3 in this case.

Solving with Equations

Percent problems can be solved by writing equations. An equation uses an equal sign (=) to show that two mathematical expressions have the same value.

Percents are fractions, and just like fractions, when finding a percent (or fraction, or portion) of another amount, you multiply.

The percent of the base is the amount.

Percent of the Base is the Amount.

\[\ \text { Percent } {\color{red}\cdot}\text { Base }{\color{blue}=}\text { Amount } \nonumber \]

In the examples below, the unknown is represented by the letter \(\ n\). The unknown can be represented by any letter or a box \(\ \square\) or even a question mark.

Write an equation that represents the following problem.

\(\ 20 \% \cdot n=30\)

Once you have an equation, you can solve it and find the unknown value. To do this, think about the relationship between multiplication and division. Look at the pairs of multiplication and division facts below, and look for a pattern in each row.

Multiplication and division are inverse operations. What one does to a number, the other “undoes.”

When you have an equation such as \(\ 20 \% \cdot n=30\), you can divide 30 by 20% to find the unknown: \(\ n=30 \div 20 \%\).

You can solve this by writing the percent as a decimal or fraction and then dividing.

\(\ n=30 \div 20 \%=30 \div 0.20=150\)

What percent of 72 is 9?

\(\ 12.5 \% \text { of } 72 \text { is } 9\).

You can estimate to see if the answer is reasonable. Use 10% and 20%, numbers close to 12.5%, to see if they get you close to the answer.

\(\ 10 \% \text { of } 72=0.1 \cdot 72=7.2\)

\(\ 20 \% \text { of } 72=0.2 \cdot 72=14.4\)

Notice that 9 is between 7.2 and 14.4, so 12.5% is reasonable since it is between 10% and 20%.

What is 110% of 24?

\(\ 26.4 \text { is } 110 \% \text { of } 24\).

This problem is a little easier to estimate. 100% of 24 is 24. And 110% is a little bit more than 24. So, 26.4 is a reasonable answer.

18 is what percent of 48?

  • \(\ 0.375 \%\)
  • \(\ 8.64 \%\)
  • \(\ 37.5 \%\)
  • \(\ 864 \%\)

Incorrect. You may have calculated properly, but you forgot to move the decimal point when you rewrote your answer as a percent. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Incorrect. You may have used \(\ 18\) or \(\ 48\) as the percent, rather than the amount or base. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Correct. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives \(\ 37.5 \%\).

Incorrect. You probably used 18 or 48 as the percent, rather than the amount or base, and also forgot to rewrite the percent as a decimal before multiplying. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Using Proportions to Solve Percent Problems

Percent problems can also be solved by writing a proportion. A proportion is an equation that sets two ratios or fractions equal to each other. With percent problems, one of the ratios is the percent, written as \(\ \frac{n}{100}\). The other ratio is the amount to the base.

\(\ \text { Percent }=\frac{\text { amount }}{\text { base }}\)

Write a proportion to find the answer to the following question.

30 is 20% of 150.

18 is 125% of what number?

  • \(\ 0.144\)
  • \(\ 694 \frac{4}{9}\) (or about \(\ 694.4\))

Incorrect. You probably didn’t write a proportion and just divided 18 by 125. Or, you incorrectly set up one fraction as \(\ \frac{18}{125}\) and set this equal to the base, \(\ n\). The percent in this case is 125%, so one fraction in the proportion should be \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Correct. The percent in this case is 125%, so one fraction in the proportion should be \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Incorrect. You probably put the amount (18) over 100 in the proportion, rather than the percent (125). Perhaps you thought 18 was the percent and 125 was the base. The correct percent fraction for the proportion is \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Incorrect. You probably confused the amount (18) with the percent (125) when you set up the proportion. The correct percent fraction for the proportion is \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Let’s go back to the problem that was posed at the beginning. You can now solve this problem as shown in the following example.

Jeff has a coupon at the Guitar Store for 15% off any purchase of $100 or more. He wants to buy a used guitar that has a price tag of $220 on it. Jeff wonders how much money the coupon will take off of the $220 original price .

The coupon will take $33 off the original price.

You can estimate to see if the answer is reasonable. Since 15% is half way between 10% and 20%, find these numbers.

\(\ \begin{array}{l} 10 \% \text { of } 220=0.1 \cdot 220=22 \\ 20 \% \text { of } 220=0.2 \cdot 220=44 \end{array}\)

The answer, 33, is between 22 and 44. So $33 seems reasonable.

There are many other situations that involve percents. Below are just a few.

Evelyn bought some books at the local bookstore. Her total bill was $31.50, which included 5% tax. How much did the books cost before tax?

The books cost $30 before tax.

Susana worked 20 hours at her job last week. This week, she worked 35 hours. In terms of a percent, how much more did she work this week than last week?

Since 35 is 175% of 20, Susana worked 75% more this week than she did last week. (You can think of this as, “Susana worked 100% of the hours she worked last week, as well as 75% more.”)

Percent problems have three parts: the percent, the base (or whole), and the amount. Any of those parts may be the unknown value to be found. To solve percent problems, you can use the equation, \(\ \text { Percent } \cdot \text { Base }=\text { Amount }\), and solve for the unknown numbers. Or, you can set up the proportion, \(\ \text { Percent }=\frac{\text { amount }}{\text { base }}\), where the percent is a ratio of a number to 100. You can then use cross multiplication to solve the proportion.

Module 7: Percents

Writing and solving percent proportions, learning outcomes.

  • Translate a statement to a proportion
  • Solve a percent proportion

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.

For example, [latex]\text{60%}={\Large\frac{60}{100}}[/latex] and we can simplify [latex]{\Large\frac{60}{100}}={\Large\frac{3}{5}}[/latex]. Since the equation [latex]{\Large\frac{60}{100}}={\Large\frac{3}{5}}[/latex] shows a percent equal to an equivalent ratio, we call it a percent proportion.

Using the vocabulary we used earlier:

[latex]{\Large\frac{\text{amount}}{\text{base}}}={\Large\frac{\text{percent}}{100}}[/latex] [latex]{\Large\frac{3}{5}}={\Large\frac{60}{100}}[/latex]

Percent Proportion

The amount is to the base as the percent is to [latex]100[/latex].

[latex]{\Large\frac{\text{amount}}{\text{base}}}={\Large\frac{\text{percent}}{100}}[/latex]

If we restate the problem in the words of a proportion, it may be easier to set up the proportion:

The amount is to the base as the percent is to one hundred.

We could also say:

The amount out of the base is the same as the percent out of one hundred.

First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

Translate to a proportion. What number is [latex]\text{75%}[/latex] of [latex]90[/latex]?

Solution If you look for the word “of”, it may help you identify the base.

Translate to a proportion. [latex]19[/latex] is [latex]\text{25%}[/latex] of what number?

Translate to a proportion. What percent of [latex]27[/latex] is [latex]9[/latex]?

Now that we have written percent equations as proportions, we are ready to solve the equations.

Translate and solve using proportions: What number is [latex]\text{45%}[/latex] of [latex]80[/latex]?

The following video shows a similar example of how to solve a percent proportion.

In the next example, the percent is more than [latex]100[/latex], which is more than one whole. So the unknown number will be more than the base.

Translate and solve using proportions: [latex]\text{125%}[/latex] of [latex]25[/latex] is what number?

Percents with decimals and money are also used in proportions.

Translate and solve: [latex]\text{6.5%}[/latex] of what number is [latex]\text{\$1.56}[/latex]?

In the following video we show a similar problem, note the different wording that results in the same equation.

Translate and solve using proportions: What percent of [latex]72[/latex] is [latex]9?[/latex]

Watch the following video to see a similar problem.

  • Question ID 146843, 146840, 146839, 146828. Authored by : Lumen Learning. License : CC BY: Attribution
  • Example 2: Solve a Percent Problem Using a Percent Proportion. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/wsBhmrmumJo . License : CC BY: Attribution
  • Example 3: Determine What Percent One Number is of Another Using a Percent Proportion. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/1GTPRROi1tE . License : CC BY: Attribution
  • Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Footer Logo Lumen Waymaker

  • 6.5 Solve Proportions and their Applications
  • Introduction
  • 1.1 Introduction to Whole Numbers
  • 1.2 Add Whole Numbers
  • 1.3 Subtract Whole Numbers
  • 1.4 Multiply Whole Numbers
  • 1.5 Divide Whole Numbers
  • Key Concepts
  • Review Exercises
  • Practice Test
  • Introduction to the Language of Algebra
  • 2.1 Use the Language of Algebra
  • 2.2 Evaluate, Simplify, and Translate Expressions
  • 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
  • 2.4 Find Multiples and Factors
  • 2.5 Prime Factorization and the Least Common Multiple
  • Introduction to Integers
  • 3.1 Introduction to Integers
  • 3.2 Add Integers
  • 3.3 Subtract Integers
  • 3.4 Multiply and Divide Integers
  • 3.5 Solve Equations Using Integers; The Division Property of Equality
  • Introduction to Fractions
  • 4.1 Visualize Fractions
  • 4.2 Multiply and Divide Fractions
  • 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
  • 4.4 Add and Subtract Fractions with Common Denominators
  • 4.5 Add and Subtract Fractions with Different Denominators
  • 4.6 Add and Subtract Mixed Numbers
  • 4.7 Solve Equations with Fractions
  • Introduction to Decimals
  • 5.1 Decimals
  • 5.2 Decimal Operations
  • 5.3 Decimals and Fractions
  • 5.4 Solve Equations with Decimals
  • 5.5 Averages and Probability
  • 5.6 Ratios and Rate
  • 5.7 Simplify and Use Square Roots
  • Introduction to Percents
  • 6.1 Understand Percent
  • 6.2 Solve General Applications of Percent
  • 6.3 Solve Sales Tax, Commission, and Discount Applications
  • 6.4 Solve Simple Interest Applications
  • Introduction to the Properties of Real Numbers
  • 7.1 Rational and Irrational Numbers
  • 7.2 Commutative and Associative Properties
  • 7.3 Distributive Property
  • 7.4 Properties of Identity, Inverses, and Zero
  • 7.5 Systems of Measurement
  • Introduction to Solving Linear Equations
  • 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
  • 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
  • 8.3 Solve Equations with Variables and Constants on Both Sides
  • 8.4 Solve Equations with Fraction or Decimal Coefficients
  • 9.1 Use a Problem Solving Strategy
  • 9.2 Solve Money Applications
  • 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
  • 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
  • 9.5 Solve Geometry Applications: Circles and Irregular Figures
  • 9.6 Solve Geometry Applications: Volume and Surface Area
  • 9.7 Solve a Formula for a Specific Variable
  • Introduction to Polynomials
  • 10.1 Add and Subtract Polynomials
  • 10.2 Use Multiplication Properties of Exponents
  • 10.3 Multiply Polynomials
  • 10.4 Divide Monomials
  • 10.5 Integer Exponents and Scientific Notation
  • 10.6 Introduction to Factoring Polynomials
  • 11.1 Use the Rectangular Coordinate System
  • 11.2 Graphing Linear Equations
  • 11.3 Graphing with Intercepts
  • 11.4 Understand Slope of a Line
  • A | Cumulative Review
  • B | Powers and Roots Tables
  • C | Geometric Formulas

Learning Objectives

By the end of this section, you will be able to:

  • Use the definition of proportion
  • Solve proportions
  • Solve applications using proportions
  • Write percent equations as proportions
  • Translate and solve percent proportions

Be Prepared 6.11

Before you get started, take this readiness quiz.

Simplify: 1 3 4 . 1 3 4 . If you missed this problem, review Example 4.44 .

Be Prepared 6.12

Solve: x 4 = 20 . x 4 = 20 . If you missed this problem, review Example 4.99 .

Be Prepared 6.13

Write as a rate: Sale rode his bike 24 24 miles in 2 2 hours. If you missed this problem, review Example 5.63 .

Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion .

A proportion is an equation of the form a b = c d , a b = c d , where b ≠ 0 , d ≠ 0 . b ≠ 0 , d ≠ 0 .

The proportion states two ratios or rates are equal. The proportion is read “ a “ a is to b , b , as c c is to d ”. d ”.

The equation 1 2 = 4 8 1 2 = 4 8 is a proportion because the two fractions are equal. The proportion 1 2 = 4 8 1 2 = 4 8 is read “ 1 “ 1 is to 2 2 as 4 4 is to 8 ”. 8 ”.

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion 20 students 1 teacher = 60 students 3 teachers 20 students 1 teacher = 60 students 3 teachers we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

Example 6.40

Write each sentence as a proportion:

  • ⓐ 3 3 is to 7 7 as 15 15 is to 35 . 35 .
  • ⓑ 5 5 hits in 8 8 at bats is the same as 30 30 hits in 48 48 at-bats.
  • ⓒ $1.50 $1.50 for 6 6 ounces is equivalent to $2.25 $2.25 for 9 9 ounces.

Try It 6.79

  • ⓐ 5 5 is to 9 9 as 20 20 is to 36 . 36 .
  • ⓑ 7 7 hits in 11 11 at-bats is the same as 28 28 hits in 44 44 at-bats.
  • ⓒ $2.50 $2.50 for 8 8 ounces is equivalent to $3.75 $3.75 for 12 12 ounces.

Try It 6.80

  • ⓐ 6 6 is to 7 7 as 36 36 is to 42 . 42 .
  • ⓑ 8 8 adults for 36 36 children is the same as 12 12 adults for 54 54 children.
  • ⓒ $3.75 $3.75 for 6 6 ounces is equivalent to $2.50 $2.50 for 4 4 ounces.

Look at the proportions 1 2 = 4 8 1 2 = 4 8 and 2 3 = 6 9 . 2 3 = 6 9 . From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?

To determine if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross product because of the cross formed. If, and only if, the given proportion is true, that is, the two sides are equal, then the cross products of a proportion will be equal.

Cross Products of a Proportion

For any proportion of the form a b = c d , a b = c d , where b ≠ 0 , d ≠ 0 , b ≠ 0 , d ≠ 0 , its cross products are equal.

Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are both equal, we have a proportion.

Example 6.41

Determine whether each equation is a proportion:

  • ⓐ 4 9 = 12 28 4 9 = 12 28
  • ⓑ 17.5 37.5 = 7 15 17.5 37.5 = 7 15

To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.

Since the cross products are not equal, 28 · 4 ≠ 9 · 12 , 28 · 4 ≠ 9 · 12 , the equation is not a proportion.

Since the cross products are equal, 15 · 17.5 = 37.5 · 7 , 15 · 17.5 = 37.5 · 7 , the equation is a proportion.

Try It 6.81

  • ⓐ 7 9 = 54 72 7 9 = 54 72
  • ⓑ 24.5 45.5 = 7 13 24.5 45.5 = 7 13

Try It 6.82

  • ⓐ 8 9 = 56 73 8 9 = 56 73
  • ⓑ 28.5 52.5 = 8 15 28.5 52.5 = 8 15

Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality .

Example 6.42

Solve: x 63 = 4 7 . x 63 = 4 7 .

Try It 6.83

Solve the proportion: n 84 = 11 12 . n 84 = 11 12 .

Try It 6.84

Solve the proportion: y 96 = 13 12 . y 96 = 13 12 .

When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

Example 6.43

Solve: 144 a = 9 4 . 144 a = 9 4 .

Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.

Another method to solve this would be to multiply both sides by the LCD, 4 a . 4 a . Try it and verify that you get the same solution.

Try It 6.85

Solve the proportion: 91 b = 7 5 . 91 b = 7 5 .

Try It 6.86

Solve the proportion: 39 c = 13 8 . 39 c = 13 8 .

Example 6.44

Solve: 52 91 = −4 y . 52 91 = −4 y .

Try It 6.87

Solve the proportion: 84 98 = −6 x . 84 98 = −6 x .

Try It 6.88

Solve the proportion: −7 y = 105 135 . −7 y = 105 135 .

Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion , we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

Example 6.45

When pediatricians prescribe acetaminophen to children, they prescribe 5 5 milliliters (ml) of acetaminophen for every 25 25 pounds of the child’s weight. If Zoe weighs 80 80 pounds, how many milliliters of acetaminophen will her doctor prescribe?

You could also solve this proportion by setting the cross products equal.

Try It 6.89

Pediatricians prescribe 5 5 milliliters (ml) of acetaminophen for every 25 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 60 pounds?

Try It 6.90

For every 1 1 kilogram (kg) of a child’s weight, pediatricians prescribe 15 15 milligrams (mg) of a fever reducer. If Isabella weighs 12 12 kg, how many milligrams of the fever reducer will the pediatrician prescribe?

Example 6.46

One brand of microwave popcorn has 120 120 calories per serving. A whole bag of this popcorn has 3.5 3.5 servings. How many calories are in a whole bag of this microwave popcorn?

Try It 6.91

Marissa loves the Caramel Macchiato at the coffee shop. The 16 16 oz. medium size has 240 240 calories. How many calories will she get if she drinks the large 20 20 oz. size?

Try It 6.92

Yaneli loves Starburst candies, but wants to keep her snacks to 100 100 calories. If the candies have 160 160 calories for 8 8 pieces, how many pieces can she have in her snack?

Example 6.47

Josiah went to Mexico for spring break and changed $325 $325 dollars into Mexican pesos. At that time, the exchange rate had $1 $1 U.S. is equal to 12.54 12.54 Mexican pesos. How many Mexican pesos did he get for his trip?

Try It 6.93

Yurianna is going to Europe and wants to change $800 $800 dollars into Euros. At the current exchange rate, $1 $1 US is equal to 0.738 0.738 Euro. How many Euros will she have for her trip?

Try It 6.94

Corey and Nicole are traveling to Japan and need to exchange $600 $600 into Japanese yen. If each dollar is 94.1 94.1 yen, how many yen will they get?

Write Percent Equations As Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.

For example, 60% = 60 100 60% = 60 100 and we can simplify 60 100 = 3 5 . 60 100 = 3 5 . Since the equation 60 100 = 3 5 60 100 = 3 5 shows a percent equal to an equivalent ratio, we call it a percent proportion . Using the vocabulary we used earlier:

Percent Proportion

The amount is to the base as the percent is to 100 . 100 .

If we restate the problem in the words of a proportion, it may be easier to set up the proportion:

We could also say:

First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

Example 6.48

Translate to a proportion. What number is 75% 75% of 90 ? 90 ?

If you look for the word "of", it may help you identify the base.

Try It 6.95

Translate to a proportion: What number is 60% 60% of 105 ? 105 ?

Try It 6.96

Translate to a proportion: What number is 40% 40% of 85 ? 85 ?

Example 6.49

Translate to a proportion. 19 19 is 25% 25% of what number?

Try It 6.97

Translate to a proportion: 36 36 is 25% 25% of what number?

Try It 6.98

Translate to a proportion: 27 27 is 36% 36% of what number?

Example 6.50

Translate to a proportion. What percent of 27 27 is 9 ? 9 ?

Try It 6.99

Translate to a proportion: What percent of 52 52 is 39 ? 39 ?

Try It 6.100

Translate to a proportion: What percent of 92 92 is 23 ? 23 ?

Translate and Solve Percent Proportions

Now that we have written percent equations as proportions, we are ready to solve the equations.

Example 6.51

Translate and solve using proportions: What number is 45% 45% of 80 ? 80 ?

Try It 6.101

Translate and solve using proportions: What number is 65% 65% of 40 ? 40 ?

Try It 6.102

Translate and solve using proportions: What number is 85% 85% of 40 ? 40 ?

In the next example, the percent is more than 100 , 100 , which is more than one whole. So the unknown number will be more than the base.

Example 6.52

Translate and solve using proportions: 125% 125% of 25 25 is what number?

Try It 6.103

Translate and solve using proportions: 125% 125% of 64 64 is what number?

Try It 6.104

Translate and solve using proportions: 175% 175% of 84 84 is what number?

Percents with decimals and money are also used in proportions.

Example 6.53

Translate and solve: 6.5% 6.5% of what number is $1.56 ? $1.56 ?

Try It 6.105

Translate and solve using proportions: 8.5% 8.5% of what number is $3.23 ? $3.23 ?

Try It 6.106

Translate and solve using proportions: 7.25% 7.25% of what number is $4.64 ? $4.64 ?

Example 6.54

Translate and solve using proportions: What percent of 72 72 is 9 ? 9 ?

Try It 6.107

Translate and solve using proportions: What percent of 72 72 is 27 ? 27 ?

Try It 6.108

Translate and solve using proportions: What percent of 92 92 is 23 ? 23 ?

Practice Makes Perfect

In the following exercises, write each sentence as a proportion.

4 4 is to 15 15 as 36 36 is to 135 . 135 .

7 7 is to 9 9 as 35 35 is to 45 . 45 .

12 12 is to 5 5 as 96 96 is to 40 . 40 .

15 15 is to 8 8 as 75 75 is to 40 . 40 .

5 5 wins in 7 7 games is the same as 115 115 wins in 161 161 games.

4 4 wins in 9 9 games is the same as 36 36 wins in 81 81 games.

8 8 campers to 1 1 counselor is the same as 48 48 campers to 6 6 counselors.

6 6 campers to 1 1 counselor is the same as 48 48 campers to 8 8 counselors.

$9.36 $9.36 for 18 18 ounces is the same as $2.60 $2.60 for 5 5 ounces.

$3.92 $3.92 for 8 8 ounces is the same as $1.47 $1.47 for 3 3 ounces.

$18.04 $18.04 for 11 11 pounds is the same as $4.92 $4.92 for 3 3 pounds.

$12.42 $12.42 for 27 27 pounds is the same as $5.52 $5.52 for 12 12 pounds.

In the following exercises, determine whether each equation is a proportion.

7 15 = 56 120 7 15 = 56 120

5 12 = 45 108 5 12 = 45 108

11 6 = 21 16 11 6 = 21 16

9 4 = 39 34 9 4 = 39 34

12 18 = 4.99 7.56 12 18 = 4.99 7.56

9 16 = 2.16 3.89 9 16 = 2.16 3.89

13.5 8.5 = 31.05 19.55 13.5 8.5 = 31.05 19.55

10.1 8.4 = 3.03 2.52 10.1 8.4 = 3.03 2.52

In the following exercises, solve each proportion.

x 56 = 7 8 x 56 = 7 8

n 91 = 8 13 n 91 = 8 13

49 63 = z 9 49 63 = z 9

56 72 = y 9 56 72 = y 9

5 a = 65 117 5 a = 65 117

4 b = 64 144 4 b = 64 144

98 154 = −7 p 98 154 = −7 p

72 156 = −6 q 72 156 = −6 q

a −8 = −42 48 a −8 = −42 48

b −7 = −30 42 b −7 = −30 42

2.6 3.9 = c 3 2.6 3.9 = c 3

2.7 3.6 = d 4 2.7 3.6 = d 4

2.7 j = 0.9 0.2 2.7 j = 0.9 0.2

2.8 k = 2.1 1.5 2.8 k = 2.1 1.5

1 2 1 = m 8 1 2 1 = m 8

1 3 3 = 9 n 1 3 3 = 9 n

In the following exercises, solve the proportion problem.

Pediatricians prescribe 5 5 milliliters (ml) of acetaminophen for every 25 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 45 45 pounds?

Brianna, who weighs 6 6 kg, just received her shots and needs a pain killer. The pain killer is prescribed for children at 15 15 milligrams (mg) for every 1 1 kilogram (kg) of the child’s weight. How many milligrams will the doctor prescribe?

At the gym, Carol takes her pulse for 10 10 sec and counts 19 19 beats. How many beats per minute is this? Has Carol met her target heart rate of 140 140 beats per minute?

Kevin wants to keep his heart rate at 160 160 beats per minute while training. During his workout he counts 27 27 beats in 10 10 seconds. How many beats per minute is this? Has Kevin met his target heart rate?

A new energy drink advertises 106 106 calories for 8 8 ounces. How many calories are in 12 12 ounces of the drink?

One 12 12 ounce can of soda has 150 150 calories. If Josiah drinks the big 32 32 ounce size from the local mini-mart, how many calories does he get?

Karen eats 1 2 1 2 cup of oatmeal that counts for 2 2 points on her weight loss program. Her husband, Joe, can have 3 3 points of oatmeal for breakfast. How much oatmeal can he have?

An oatmeal cookie recipe calls for 1 2 1 2 cup of butter to make 4 4 dozen cookies. Hilda needs to make 10 10 dozen cookies for the bake sale. How many cups of butter will she need?

Janice is traveling to Canada and will change $250 $250 US dollars into Canadian dollars. At the current exchange rate, $1 $1 US is equal to $1.01 $1.01 Canadian. How many Canadian dollars will she get for her trip?

Todd is traveling to Mexico and needs to exchange $450 $450 into Mexican pesos. If each dollar is worth 12.29 12.29 pesos, how many pesos will he get for his trip?

Steve changed $600 $600 into 480 480 Euros. How many Euros did he receive per US dollar?

Martha changed $350 $350 US into 385 385 Australian dollars. How many Australian dollars did she receive per US dollar?

At the laundromat, Lucy changed $12.00 $12.00 into quarters. How many quarters did she get?

When she arrived at a casino, Gerty changed $20 $20 into nickels. How many nickels did she get?

Jesse’s car gets 30 30 miles per gallon of gas. If Las Vegas is 285 285 miles away, how many gallons of gas are needed to get there and then home? If gas is $3.09 $3.09 per gallon, what is the total cost of the gas for the trip?

Danny wants to drive to Phoenix to see his grandfather. Phoenix is 370 370 miles from Danny’s home and his car gets 18.5 18.5 miles per gallon. How many gallons of gas will Danny need to get to and from Phoenix? If gas is $3.19 $3.19 per gallon, what is the total cost for the gas to drive to see his grandfather?

Hugh leaves early one morning to drive from his home in Chicago to go to Mount Rushmore, 812 812 miles away. After 3 3 hours, he has gone 190 190 miles. At that rate, how long will the whole drive take?

Kelly leaves her home in Seattle to drive to Spokane, a distance of 280 280 miles. After 2 2 hours, she has gone 152 152 miles. At that rate, how long will the whole drive take?

Phil wants to fertilize his lawn. Each bag of fertilizer covers about 4,000 4,000 square feet of lawn. Phil’s lawn is approximately 13,500 13,500 square feet. How many bags of fertilizer will he have to buy?

April wants to paint the exterior of her house. One gallon of paint covers about 350 350 square feet, and the exterior of the house measures approximately 2000 2000 square feet. How many gallons of paint will she have to buy?

Write Percent Equations as Proportions

In the following exercises, translate to a proportion.

What number is 35% 35% of 250 ? 250 ?

What number is 75% 75% of 920 ? 920 ?

What number is 110% 110% of 47 ? 47 ?

What number is 150% 150% of 64 ? 64 ?

45 45 is 30% 30% of what number?

25 25 is 80% 80% of what number?

90 90 is 150% 150% of what number?

77 77 is 110% 110% of what number?

What percent of 85 85 is 17 ? 17 ?

What percent of 92 92 is 46 ? 46 ?

What percent of 260 260 is 340 ? 340 ?

What percent of 180 180 is 220 ? 220 ?

In the following exercises, translate and solve using proportions.

What number is 65% 65% of 180 ? 180 ?

What number is 55% 55% of 300 ? 300 ?

18% 18% of 92 92 is what number?

22% 22% of 74 74 is what number?

175% 175% of 26 26 is what number?

250% 250% of 61 61 is what number?

What is 300% 300% of 488 ? 488 ?

What is 500% 500% of 315 ? 315 ?

17% 17% of what number is $7.65 ? $7.65 ?

19% 19% of what number is $6.46 ? $6.46 ?

$13.53 $13.53 is 8.25% 8.25% of what number?

$18.12 $18.12 is 7.55% 7.55% of what number?

What percent of 56 56 is 14 ? 14 ?

What percent of 80 80 is 28 ? 28 ?

What percent of 96 96 is 12 ? 12 ?

What percent of 120 120 is 27 ? 27 ?

Everyday Math

Mixing a concentrate Sam bought a large bottle of concentrated cleaning solution at the warehouse store. He must mix the concentrate with water to make a solution for washing his windows. The directions tell him to mix 3 3 ounces of concentrate with 5 5 ounces of water. If he puts 12 12 ounces of concentrate in a bucket, how many ounces of water should he add? How many ounces of the solution will he have altogether?

Mixing a concentrate Travis is going to wash his car. The directions on the bottle of car wash concentrate say to mix 2 2 ounces of concentrate with 15 15 ounces of water. If Travis puts 6 6 ounces of concentrate in a bucket, how much water must he mix with the concentrate?

Writing Exercises

To solve “what number is 45% 45% of 350 ” 350 ” do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

To solve “what percent of 125 125 is 25 ” 25 ” do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

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  • Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
  • Publisher/website: OpenStax
  • Book title: Prealgebra 2e
  • Publication date: Mar 11, 2020
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/prealgebra-2e/pages/1-introduction
  • Section URL: https://openstax.org/books/prealgebra-2e/pages/6-5-solve-proportions-and-their-applications

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Proportions

Proportion says that two ratios (or fractions) are equal.

We see that 1-out-of-3 is equal to 2-out-of-6

The ratios are the same, so they are in proportion.

Example: Rope

A rope's length and weight are in proportion.

When 20m of rope weighs 1kg , then:

  • 40m of that rope weighs 2kg
  • 200m of that rope weighs 10kg

20 1 = 40 2

When shapes are "in proportion" their relative sizes are the same.

Example: International paper sizes (like A3, A4, A5, etc) all have the same proportions:

So any artwork or document can be resized to fit on any sheet. Very neat.

Working With Proportions

NOW, how do we use this?

Example: you want to draw the dog's head ... how long should it be?

Let us write the proportion with the help of the 10/20 ratio from above:

? 42 = 10 20

Now we solve it using a special method:

Multiply across the known corners, then divide by the third number

And we get this:

? = (42 × 10) / 20 = 420 / 20 = 21

So you should draw the head 21 long.

Using Proportions to Solve Percents

A percent is actually a ratio! Saying "25%" is actually saying "25 per 100":

25% = 25 100

We can use proportions to solve questions involving percents.

The trick is to put what we know into this form:

Part Whole = Percent 100

Example: what is 25% of 160 ?

The percent is 25, the whole is 160, and we want to find the "part":

Part 160 = 25 100

Multiply across the known corners, then divide by the third number:

Part = (160 × 25) / 100 = 4000 / 100 = 40

Answer: 25% of 160 is 40.

Note: we could have also solved this by doing the divide first, like this:

Part = 160 × (25 / 100) = 160 × 0.25 = 40

Either method works fine.

We can also find a Percent:

Example: what is $12 as a percent of $80 ?

Fill in what we know:

$12 $80 = Percent 100

Multiply across the known corners, then divide by the third number. This time the known corners are top left and bottom right:

Percent = ($12 × 100) / $80 = 1200 / 80 = 15%

Answer: $12 is 15% of $80

Or find the Whole:

Example: The sale price of a phone was $150, which was only 80% of normal price. What was the normal price?

$150 Whole = 80 100

Whole = ($150 × 100) / 80 = 15000 / 80 = 187.50

Answer: the phone's normal price was $187.50

Using Proportions to Solve Triangles

We can use proportions to solve similar triangles.

Example: How tall is the Tree?

Sam tried using a ladder, tape measure, ropes and various other things, but still couldn't work out how tall the tree was.

proportion tree

But then Sam has a clever idea ... similar triangles!

Sam measures a stick and its shadow (in meters), and also the shadow of the tree, and this is what he gets:

Now Sam makes a sketch of the triangles, and writes down the "Height to Length" ratio for both triangles:

Height: Shadow Length:     h 2.9 m = 2.4 m 1.3 m

h = (2.9 × 2.4) / 1.3 = 6.96 / 1.3 = 5.4 m (to nearest 0.1)

Answer: the tree is 5.4 m tall.

And he didn't even need a ladder!

The "Height" could have been at the bottom, so long as it was on the bottom for BOTH ratios, like this:

Let us try the ratio of "Shadow Length to Height":

Shadow Length: Height:     2.9 m h = 1.3 m 2.4 m

It is the same calculation as before.

A "Concrete" Example

Ratios can have more than two numbers !

For example concrete is made by mixing cement, sand, stones and water.

concrete pouring

A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6 .

We can multiply all values by the same amount and still have the same ratio.

10:20:60 is the same as 1:2:6

So when we use 10 buckets of cement, we should use 20 of sand and 60 of stones.

Example: you have just put 12 buckets of stones into a mixer, how much cement and how much sand should you add to make a 1:2:6 mix?

Let us lay it out in a table to make it clearer:

You have 12 buckets of stones but the ratio says 6.

That is OK, you simply have twice as many stones as the number in the ratio ... so you need twice as much of everything to keep the ratio.

Here is the solution:

And the ratio 2:4:12 is the same as 1:2:6 (because they show the same relative sizes)

So the answer is: add 2 buckets of Cement and 4 buckets of Sand. (You will also need water and a lot of stirring....)

Why are they the same ratio? Well, the 1:2:6 ratio says to have :

  • twice as much Sand as Cement ( 1 : 2 :6)
  • 6 times as much Stones as Cement ( 1 :2: 6 )

In our mix we have:

  • twice as much Sand as Cement ( 2 : 4 :12)
  • 6 times as much Stones as Cement ( 2 :4: 12 )

So it should be just right!

That is the good thing about ratios. You can make the amounts bigger or smaller and so long as the relative sizes are the same then the ratio is the same.

Percentages, Ratios, and Proportions

This section covers:

Percentages and Percent Changes

Ratios and proportions, unit multipliers, using percentages with ratios.

  • More Practice

Note :  For more problems with percents and ratios, see the Algebra Word Problems section .

Percentages are something you are probably quite familiar with because of your shopping habits, right? How many times have you been to the store when everything is 20% off? Do you notice how many people around you (adults, usually!) have no idea how to figure out what the sale price is? The easiest example of percentages is 50% off, which means that the item is half price.

Percentages really aren’t that difficult if you truly understand what they are. The word “percentage” comes from the word “per cent”, which means “per hundred” in Latin.  Remember that “per” usually means “over”. So “per cent” literally means “over 100 ” or “divided by 100 “. And remember what “of” typically means? I’ll write it again, since it’s so important:

OF  =  TIMES

When we say “ 20% off of something”, let’s translate it to “ 20 over (or divided by) 100 — then times the original price”, and that will be the amount we subtract from the original price.

Remember that we cannot use a percentage in math;  we need to turn it into a decimal.   To turn a percentage into a decimal, we move the decimal 2 places to the left (because we need to divide by 100 ), and if we need to turn a decimal back into a percentage, we move the decimal 2 places to the  right (because we need to multiply by 100 ).

I like to think of it this way : When we’re taking away the %, we are afraid of it, so we move 2 decimal places away from it (or to the left ). When we need to turn a number into a %, we like it, so we move 2 decimals towards it (or to the right ).

Let’s get back to our percentage example.  If there’s a dress we like for say $50 , and it’s 20% off (“off” means take-away or minus!), we’ll do the math to figure out the sales price.  This is called a percent change   problem.

Amount of sale : $ \displaystyle 20\%\,\,\text{of }\,\$50=.2\times \$50=\$10.\,\,\,\,\$50-\$10=\$40$. The dress would be $40 .

(See how we had to turn the 20% into a decimal by taking away the % sign and moving 2 decimals to the left, or away from it, since we didn’t like it?)

We could have also multiplied the original price by $ \displaystyle 80\%\,(100\%-20\%)$, or $ \displaystyle \frac{{80}}{{100}}$, since that’s what we’ll be paying if we get 20% off ( 100% full price minus 20% discount equals 80% discounted price):

Price of discounted dress : $ \displaystyle 80\%\,\,\text{of }\,\$50=.8\times \$50=\$40$. This method has fewer steps.

This shopping example is a percent decrease problem; the following is the formula for that. Make sure you relate this formula back to the example above.

$ \displaystyle \text{Newer}\,\,\text{lower}\,\,\text{price =}\,\,\text{original}\,\,\text{price}\,\,-\,\,\left( {\text{original}\,\,\text{price}\,\,\times \,\,\left. {\frac{{\text{percentage}\,\,\text{off}}}{{100}}} \right)} \right.$

$ \displaystyle \$50-\left( {\$50\,\,\times \,\,\left. {\frac{{20}}{{100}}} \right)} \right.\,\,=\,\,\$50-\$10=\$40$

Notice that we worked the math in the parentheses first (we will get to this in more detail later) .

Now let’s talk about a percent increase problem, which is also a percent change problem. A great example of a percent increase is the tax you pay on this dress. Tax is a percentage (usually) that you add on to what you pay so we can continue driving on the streets free and going to public school free.

If we need to add on 8.25% sales tax to the $40 that we are going to spend on the dress, we’ll have to know the percent increase formula, but let’s first figure it out without the formula.  Tax is the amount we have to add that is based on a percentage of the price that we’re paying for the dress.

The tax would be 8.25% or .0825 (remember – we don’t like the %, so we take it away and move away from it?) times the price of the dress and then add it back to the price of the dress.

Total price with tax:   $ \displaystyle \$50+(8.25\%\times 50)=\$50+(.0825\times 50)=\$50+\$4.125=\$54.125=\$54.13$.

Note that we rounded up to two decimal places, since we’re dealing with money. Note also that we did the math inside the parentheses first.

The total price of the dress would be $54.13 .

Here’s the formula:

$ \displaystyle \text{Price}\,\,\text{with}\,\,\text{tax}=\,\,\text{original}\,\,\text{price}+\,\,\left( {\text{original}\,\,\text{price}\,\,\times \,\,\left. {\frac{{\text{tax}\,\,\text{percentage}}}{{100}}} \right)} \right.$

$ \displaystyle \$50+\left( {\$50\,\,\times \,\,\left. {\frac{{8.25}}{{100}}} \right)} \right.\,\,=\,\,\$50+\left( {\$50\times \left. {.0825} \right)} \right.\,\,=\,\,\$50+\$4.125=\$54.125=\$54.13$

Another way we can figure percent increase is to just multiply the original amount by 1 (to make sure we include it) and also multiply it by the tax rate and add them together (this is actually using something called distributing, which we’ll talk about in Algebra):

$ \displaystyle \text{Price}\,\,\text{with}\,\,\text{tax}\,\,\text{=}\,\,\text{original}\,\,\text{price}\,\,\times \,\,\left( {1+\left. {\frac{{\text{tax}\,\,\text{percentage}}}{{100}}} \right)} \right.$

$ \displaystyle \$50\,\times \,\left( {1+\left. {\frac{{8.25}}{{100}}} \right)} \right.=\$50\,\times \,\left( {1+\left. {.0825} \right)} \right.=\$50\times 1.0825=\$54.125=\$54.13$

If we need to figure out the actual percent decrease or increase ( percent change ), we can use the following formula:

$ \displaystyle \text{Percent Increase}=\frac{{\text{New Price}-\text{Old Price}}}{{\text{Old Price}}}\,\times 100$

$ \displaystyle \text{Percent Decrease}\,=\frac{{\text{Old Price}-\text{New Price}}}{{\text{Old Price}}}\,\,\times \,100$

For example, say we want to work backwards to get the percentage of sales tax that we pay (percent increase). If we know that the original (old) price is $50 , and the price we pay (new price) is $54.13 , we could get the % we pay in tax this way  (note that since we rounded to get the 54.13 , our answer is off a little):

$ \displaystyle \text{Percent Increase (Tax)}\,\,=\frac{{54.13-50}}{{50}}\,\,\times \,100\,=\,8.26%$

Sometimes we have to work a little backwards in the problem to get the right answer. For example, we may have a problem that says something like this:

Your favorite pair of shoes are on sale for 30% off. The sale price is $62.30 . What was the original price?

To do this problem, we have to think about the fact that if the shoes are on sale for 30% , we need to pay 70% for them. Also remember that “of = times”. We can set it up this way:

$ \displaystyle \,\,.7\,\,\times \,\,?=\$62.30\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,? = \frac{{62.30}}{{.7}} = \$89$

The original price of the shoes would have been $89 before tax.

In the Algebra sections, we will address solving the following types of percentage problems, but I’ll briefly address them here if you need to do them now. If you don’t totally follow how to get the answers, don’t worry about it, since we’ll cover “word problems” later!

One other way to address percentages is the “$ \displaystyle \frac{{\text{is}}}{{\text{of}}}$” trick, which we’ll address below.

Ratios  are just a comparison of two numbers. They look a little scary since they involve fractions, but they really aren’t bad at all. Again, they are typically used when you are comparing two things — like cost of one pair of shoes to another pair, or maybe even the number of shirts you have compared to the number of jeans you have.

Let’s use that as an example. Let’s say you have about 5 shirts for every one pair of jeans you have, and you figure this same ratiois pretty typical among your friends. You can write your ratio in a fraction like $ \displaystyle \frac{5}{1}$, or you can use a colon in between the two numbers, like  5 : 1 (spoken as “ 5 to 1 ”). The fractions over 1 is actually a rate  (this word is related to the word ratio!), for example, just like when you think of miles per hour. Our rate is shirts per one pair of jeans – 5 shirts for every pair of jeans.

Also note that this particular ratio is a unit rate , since the second number (denominator in the fraction) is 1 .

Let’s say you know your friend Alicia has 7 pairs of jeans and you’re wondering how many shirts she has, based on the ratio or rate of 5 shirts to one pair of jeans.  We can do this with math quite easily by setting up the following proportion , which is an equation (setting two things equal to one another) with a ratio on each side:

How do we figure out how many shirts Alicia has? One way is just to think about reducing or expanding fractions. Let’s expand the fraction $ \displaystyle \frac{5}{1}$ to another fraction that has 7 on the bottom:

$ \displaystyle \frac{{\text{shirts}}}{{\text{jeans}}}\,=\,\frac{5}{1}\,=\,\frac{5}{1}\,\times \,1\,=\frac{5}{1}\,\,\times \,\,\frac{7}{7}\,=\frac{{35}}{7}$

Alicia would have about 35 shirts.

Now I’m going to also show you a concept called cross-multiplying , which is very, very useful, even when we get into Algebra, Geometry and up through Calculus! This is a much easier way to do these types of problems.

Remember the “butterfly up” concept when we’re comparing fractions, and remember how the fractions are equal when the “butterfly up” products are equal?

We’re going to use this concept to set the fractions or ratios equal so we know how many shirts Alicia has:

We know that 5 × 7 = 35 , so we need to know what multiplied by 1 will give us 35 .  35 !! Alicia has 35 shirts!!! See how easy that was?  Now if we didn’t have the 1 as a factor to get to 35 , we’d have to divide 35 by the number under the 5 to get the answer.  This is because dividing “undoes” multiplying.

One of my students also suggested to use the “ WON ” method for proportions. To do this, you set up a table with WON at the top.  “W” stands for Words , “O” stands for Original or Old , and “N” stands for New (in this example, for Alicia). Put the words and numbers in the table, and then cross multiply like we did earlier. Again, we get that Alicia has 35 shirts , based on my proportion of 5 shirts to every pair of jeans, and the fact that she has 7 pairs of jeans.

Let’s try a cooking example with proportions, since sometimes the recipe might give you the amounts in tablespoons, for example, and you only have a measuring spoon with teaspoons. We know from the Fractions section that 1 tablespoon = 3 teaspoons, and let’s say the recipe calls for 2 tablespoons. This seems pretty easy to do without the proportion, but let’s set it up anyway, so you can see how easy it is to use proportions:

$ \require{cancel} \displaystyle \frac{{\text{teaspoons}}}{{\text{tablespoons}}}\,\,\,\,\,={}^{6}{{\xcancel{{\frac{3}{1}\,\,\,=\,\,\,\frac{?}{2}}}}^{6}}$

Now let’s go on to a more complicated example that relates back to converting numbers back and forth between the metric system and our customary system here.  (For more discussion on the metric system, see the  Metric System  section).

Let’s say we have 13 meters of something and we want to know how many feet this is.  We can either look up how many feet are in 1 meter, or how many meters are in 1 foot – it really doesn’t matter – but we need a conversion number.

We find that 1 meter equals approximately 3.28 feet. Let’s set all this up in a proportion. Remember to keep the same unit either on the tops of the proportion, or on the sides – it works both ways:

$ \displaystyle \frac{{\text{meters}}}{{\text{meters}}}\,\,=\,\,\frac{{\text{feet}}}{{\text{feet}}}\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\frac{{\text{meters}}}{{\text{feet}}}\,\,=\,\,\frac{{\text{meters}}}{{\text{feet}}}\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\frac{{\text{feet}}}{{\text{meters}}}\,\,=\,\,\frac{{\text{feet}}}{{\text{meters}}}$

Let’s solve both two different ways to get the number of feet in 13 meters. Notice that we can turn proportions sideways, move the “ = ” sideways too, and solve – this is sort of how we got from the first equation to the second above.

Here’s an example where we have to do some dividing with our cross multiplying. Try to really understand why we have to divide by 2 to get the answer (it “undoes” the multiplying):

$ \displaystyle \frac{5}{2}\,=\,\frac{?}{9}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5\,\,\times \,\,9\,=2\,\times \,?\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,?\,=\,\frac{{5\,\times \,9}}{2}\,=\,\frac{{45}}{2}\,=\,\,22\frac{1}{2}$

We can also use what we call unit multipliers  to change numbers from one unit to another.  The idea is to multiply fractions to get rid of the units we don’t want. You probably will use this technique some day when you take Chemistry; it may be called Dimensional Analysis .

Let’s say we want to use two unit multipliers to convert 58 inches to yards.

Since we have inches and we want to end up with yards, we’ll multiply by ratios (fractions) that relate the units to each other. We can do this because we are really multiplying by “ 1 ”, since the top and bottom amounts will be the same (just the units will be different). Let’s first set this up with the units we have to see what we’ll need to have on the top and the bottom. I put 1 ’s under the first and last items to make them look like fractions:

$ \displaystyle \frac{{58\text{ inches}}}{1}\,\,\times \,\,\frac{?}{?}\,\,\times \,\,\frac{?}{?}\,\,=\,\,\frac{{?\text{ yards}}}{1}$

We need to get rid of the inches unit on the top and somehow get the yards unit on the top; since the problem calls for 2 unit multipliers, we’ll include feet to do this:

$ \require{cancel} \displaystyle \frac{{58\text{ }\cancel{{\text{inches}}}}}{1}\,\times \,\frac{{?\text{ }\cancel{{\text{feet}}}}}{{?\text{ }\cancel{{\text{inches}}}}}\,\times \,\frac{{?\text{ }\,\text{yards}}}{{?\text{ }\cancel{{\text{feet}}}}}\,=\,\frac{{\text{? }\,\text{yards}}}{\text{1}}$

Now just fill in how many inches are in a foot, and how many feet are in a yard, and we can get the answer with real numbers:

$ \displaystyle \frac{{58\text{ }\cancel{{\text{inches}}}}}{1}\,\times \,\frac{{1\text{ }\cancel{{\text{foot}}}}}{{12\text{ }\cancel{{\text{inches}}}}}\,\times \,\frac{{1\text{ yard}}}{{3\text{ }\cancel{{\text{feet}}}}}\,=\,\frac{{58\times 1\times 1\text{ yards}}}{{1\times 12\times 3}}\,=\,\frac{{58}}{{36}}\text{ }\,\text{yards}\,=\,\frac{{29}}{{18}}\text{ }\,\text{yards}$

Here’s another example where we use two unit multipliers since we are dealing with square units:

Use two unit multipliers to convert 100 square kilometers to square meters.

$ \displaystyle \frac{{100\text{ }\cancel{{\text{kilometers}}}\times \cancel{{\text{kilometers}}}}}{1}\,\times \,\frac{{1000\text{ meters}}}{{1\text{ }\cancel{{\text{kilometer}}}}}\,\times \,\frac{{1000\text{ meters}}}{{1\text{ }\cancel{{\text{kilometer}}}}}\,=\,100,000,000\,\, \text{meter}{{\text{s}}^{2}}$

Now let’s revisit percentages and show how proportions can help with them too! One trick to use is the $ \displaystyle \frac{{\text{is}}}{{\text{of}}}$ and $ \displaystyle \frac{{\text{part}}}{{\text{whole}}}$ tricks. You can remember these since the word that comes first in the alphabet (“is” and “part”) are on the top of the fractions.

You can typically solve percentage problems by using the following formula:

$ \displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{\text{ }\!\!\%\!\!\text{ }}{{100}}$

What this means is that the number around the “is” in an equation is on top of the proportion, and the number that comes after the “of” in an equation is on bottom of the proportion, and the percentage is over the 100 .

You can also think of this as the following, but you have to remember that sometimes the part may be actually be bigger than the whole (if the percentage is greater than 100):

$ \displaystyle \frac{{\text{part}}}{{\text{whole}}}=\frac{\text{ }\!\!\%\!\!\text{ }}{{100}}$

Here are some examples, using the same problems that we did above in the Percentages section.  (Later, in the Algebra section, we’ll learn how to translate math word problems like these word-for-word from English to math.)

  • What is 20% of 100 ?   Since we know that the 20 of the % part, we put that over the 100. The 100 comes after the “of”, so we put that on the bottom. Also, we’re looking for the “part” of the “whole” here.

$ \displaystyle \frac{{\text{is}}}{{\text{of}}}=\,\frac{\%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{?}{{100}}=\frac{{20}}{{100}}\,\,\,\,\,\,\,\,\,?=20$

  • 100 is what percentage of 200 ?   The 100 is close to the “is” so we put that on the top. The 200 comes after the “of”, so we put that on the bottom. Also, we know the 100 is the “part” of the 200 .

$ \displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{100}}{{200}}=\frac{{?\,\,\,\%}}{{100}}\,\,\,\,\,\,\,\,\,?=50$

  • 200 is 50% of what number?    The 200 is close to the “is” and we don’t know what the “of” is. The 50 is the percentage. Also, 200 is the “part”, so we need to find the “whole”.

$ \displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{200}}{?}=\frac{{50}}{{100}}\,\,\,\,\,\,\,\,\,?=400$

Here are a few more problems on rates and percentages :

Remember also – if you’re not quite sure what you’re doing, think of the problems with easier numbers and see how you’re doing it!  This can help a lot of the time. Learn these rules and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the  Mathway  site, where you can register for the  full version  (steps included) of the software.  You can even get math worksheets.

You can also go to the  Mathway  site here , where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Negative Numbers and Absolute Value   – you are ready!!

IMAGES

  1. Solve Percent Problems using the Proportion Method

    solving percent problems using proportions

  2. How To Solve Percent Proportions In 12 Minutes

    solving percent problems using proportions

  3. Solving Percent Problems using Proportions (FLIP Lesson)

    solving percent problems using proportions

  4. Using Proportions to Solve Percent Problems

    solving percent problems using proportions

  5. Example 2: Solve a Percent Problem Using a Percent Proportion

    solving percent problems using proportions

  6. Percents Solving Percent Problems Using Proportions

    solving percent problems using proportions

VIDEO

  1. Solving Percent Proportions

  2. Solving Problems using Percent Proportions TEKS 6.5B 11 July 2018

  3. Lesson 105: Using Proportions to Solve Percent Problems (Saxon Math, Level 6)

  4. Solving Proportions and working with multistep percent problems

  5. Solving Percent Question 1

  6. Percent of an unknown number using a Bar Model #maths #mathtrick #shorts

COMMENTS

  1. 5.2.1: Solving Percent Problems

    Applied Mathematics Developmental Math (NROC) 5: Percents 5.2: Solving Percent Problems 5.2.1: Solving Percent Problems Expand/collapse global location 5.2.1: Solving Percent Problems Page ID The NROC Project

  2. Percent and Proportions

    For example: ____ is ____ % of ____. In the problem, 8 is what percent of 20?, the number 8 is some percent of the number 20. Looking at this problem, it is clear that 8 is the part and 20 is the whole.

  3. Writing and Solving Percent Proportions

    The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio. For example, 60% = 60 100 60% = 60 100 and we can simplify 60 100 = 3 5 60 100 = 3 5.

  4. 6.5 Solve Proportions and their Applications

    Solve proportions Solve applications using proportions Write percent equations as proportions Translate and solve percent proportions Be Prepared 6.11 Before you get started, take this readiness quiz. Simplify: 1 3 4. If you missed this problem, review Example 4.44. Be Prepared 6.12 Solve: x 4 = 20. If you missed this problem, review Example 4.99.

  5. Solving Percent Problems using Proportions (FLIP Lesson)

    Solving Percent Problems using Proportions (FLIP Lesson) - YouTube 0:00 / 13:28 Solving Percent Problems using Proportions (FLIP Lesson) Rob Oliver 1.16K subscribers Subscribe Subscribed 306...

  6. Proportions

    A percent is actually a ratio! Saying "25%" is actually saying "25 per 100": 25% = 25 100. We can use proportions to solve questions involving percents. The trick is to put what we know into this form: Part Whole = Percent 100.

  7. PDF Percent Problems: Proportion Method

    A percent equation can be solved using proportions. You may wish to review proportions in your text. The proportion must have two equal ratios. One of the ratios will be determined by the Amount percent. The other ratio will be the Amount to the Base, written .

  8. Use Proportions to Solve Percent Problems ( Read )

    a 968 = 37 100 100 a = 968 × 37 100 a = 35816. Then, divide by 100 to find the number of uniforms to purchase. 100 a = 35816 100 a 100 = 35816 100 a = 358.16. The answer is 358.16. Therefore, the team needs to purchase 358 uniforms. Since you can use a proportion to solve for any missing variable, you can also find the whole if you know a ...

  9. Solving percent problems (video)

    Questions Tips & Thanks Want to join the conversation? Sort by: Top Voted edgel26 4 years ago The way I thought of it was that you multiply 150 times 4, knowing that 25% is 1/4 of 100% soooo by doing this we would find the number that 150 would be 25% of. Is this right and did I confuse anyone? It was just the simplest way I thought of it •

  10. Multi-step ratio and percent problems (article)

    Use proportional relationships to solve multistep ratio and percent problems. Examples include simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Problem 1: Magic carpet A magic carpet is made with three colors of yarn. The ratio of each color in the carpet is shown below: 10

  11. Solving Percent Problems Using Proportions

    This video walks through several examples of how to solve basic percent problems by using proportions.

  12. Part, Whole, & Percent Proportion Word Problems

    This math video tutorial lesson explains how to solve percent proportion word problems for students in 7th grade. Percentages Made Easy: ...

  13. Percentages, Ratios, and Proportions

    Pre-Algebra Percentages, Ratios, and Proportions Percentages, Ratios, and Proportions This section covers: Percentages and Percent Changes Ratios and Proportions Unit Multipliers Using Percentages with Ratios More Practice Note: For more problems with percents and ratios, see the Algebra Word Problems section. Percentages and Percent Changes

  14. Example 1: Solve a Percent Problem Using a Percent Proportion

    Goes through the steps of solving what is 24% of 325 using a percent proportion.

  15. PDF Solving Ratio and Percent Problems Using Proportional Relationships

    You can find the amount of sales tax by using the following proportion: percent sales tax = amount of sales tax 100 amount to be taxed Let's say you bought a jacket for $85. If the sales tax is 7.5%, what is the tax? What would be the total cost of the jacket? Remember that we can only add percents to percents.

  16. Percent Proportion

    This video shows viewers how to use the percent proportion to solve percent related problems.

  17. Solving proportions (practice)

    Do 7 problems. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

  18. Percent Problems Using Proportions

    Using the bar, they should try to determine about what fraction of the whole represents the percent given. Use: 25% of 20. 40% of 30. 10% of 50. 75% of 10. After estimating using the Percent Estimator, have the students set up a proportion using part/total = %/100 as the ratios, and have them find the exact number. Q.

  19. Solving Percent Problems Using a Proportion

    In this video, we will solve a variety of questions on percentage using proportions. #algebra #algebraformulas #algebratricks #algebrahelp #math #mathvideo #...

  20. Percent Problems Using Proportions Flashcards

    Study with Quizlet and memorize flashcards containing terms like Use P/100 = A/B to solve the swallowing problem for the unknown quantity. Round your answer to the nearest tenth, if necessary. ___% of 100 is 65, ____% of 140 is 42., 89% of 732 and more.