problem solving using proportions

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How to Solve Proportions

Last Updated: March 26, 2024

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 94,181 times.

What is the "vertical" way to solve a proportion?

How can I solve a proportion with the "horizontal" method?

How do I solve a proportion step by step by cross-multiplying?

How do you find the missing value in a proportion with a table of ratios?

• Each column in this table represents a fraction. All of the fractions in this table are equal to each other.

• The two answers are the same, which means your answer is correct.

How do you solve percent proportions?

How do you solve proportions algebraically?

• You can change the left hand side of the equation, as long as you do the same math to the right hand side.

• To get rid of the fraction on the left, multiply both sides by 27:

How do you solve a proportion with a variable on both sides?

• Warning : This is a difficult example. If you haven't learned about quadratic equations yet, you might want to skip this part.

• You can now solve this as a quadratic equation , using any method that you've learned.

Proportions Calculator, Practice Problems, and Answers

Community Q&A

• The algebraic method above works with any proportion. But for a specific proportion, there is often a faster way to use algebra to find the answer. As you learn more algebra, this will get easier. Thanks Helpful 0 Not Helpful 0

You Might Also Like

• ↑ http://www.mathvillage.info/node/72
• ↑ https://www.youtube.com/watch?v=nwsDiID7UtQ
• ↑ https://www.youtube.com/watch?v=Uo8HgcyfRFI
• ↑ https://www.purplemath.com/modules/ratio2.htm

About This Article

To solve proportions, start by taking the numerator, or top number, of the fraction you know and multiplying it with the denominator, or bottom number, of the fraction you don’t know. Next, take that number and divide it by the denominator of the fraction you know. Now you can replace x with this final number. For example, to figure out “x” in the problem 3/4 = x/8, multiply 3 x 8 to get 24, then divide 24 / 4 to get 6, or the value of x. To learn how to use proportions to determine percentages, read on! Did this summary help you? Yes No

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6.5 Solve Proportions and their Applications

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 ?

Section 6.5 Exercises

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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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.

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.

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.

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Proportion word problems

It is very important to notice that if the ratio on the left is a ratio of number of liters of water to number of lemons, you have to do the same ratio on the right before you set them equal. 

More interesting proportion word problems

Check this site if you want to solve more proportion word problems.

Ratio word problems

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Real World Algebra by Edward Zaccaro

Algebra is often taught abstractly with little or no emphasis on what algebra is or how it can be used to solve real problems. Just as English can be translated into other languages, word problems can be "translated" into the math language of algebra and easily solved. Real World Algebra explains this process in an easy to understand format using cartoons and drawings. This makes self-learning easy for both the student and any teacher who never did quite understand algebra. Includes chapters on algebra and money, algebra and geometry, algebra and physics, algebra and levers and many more. Designed for children in grades 4-9 with higher math ability and interest but could be used by older students and adults as well. Contains 22 chapters with instruction and problems at three levels of difficulty.

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Solving Proportions Worksheets

Learn how to solve proportions with this set of printer-friendly worksheets that are specially designed for students of grade 6 through high school. A series of multi-level worksheets require students to solve proportions using the cross product method and the answers so derived will be in the form of whole numbers, fractions or decimals. Also, solve worksheets that contain the variables in algebraic expressions. A variety of authentic word problems that incorporate real-life scenarios are also featured here. Check out some of these handouts for free!

Solve the Proportion: Level 1

Each set of proportions rendred in these 6th grade pdfs contains a variable. Cross-multiply the ratios to determine the value of the unknown. Answers derived will comprise only whole numbers.

• Download the set

Solve the Proportion: Level 2

Use the cross product method to solve each proportion in these printable worksheets for grade 7. The value of the missing variables will be in the form of either proper fractions or mixed fractions.

Solve the Proportion - Decimals

Students in grade 8 need to determine the product of extremes and the product of means to solve the equations that contain decimals and eventually evaluate the unknown. The answer key is available in the next page of each worksheet.

Level 1: Solve the Proportion - Algebraic Expression

Evaluate the proportions involving algebraic expressions with two terms. Use the proportionality rule and solve the equations to obtain the value of the missing variable.

Level 2: Solve the Proportion - Algebraic Expression

Use the cross product rule to obtain the equation that involves the rational expression on both sides. There are eight problems in each worksheet for 8th grade and high school students.

Proportion Word Problems

Observe the proportional relationships in real world with this series of word problems pdf worksheets; set up the proportionality with the given values and solve for the unknown.

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Proportion Calculator

What is a proportion – proportion definition, constant of proportionality – values being directly and inversely proportional, how to solve proportions – proportion examples, the golden ratio, law of multiple proportions – proportional relationships in chemistry, directly proportional tax definition.

The proportion calculator helps compute identical proportions. Before we can use the calculator, it's important to understand what a proportion is and how to solve proportions by hand. To help you understand these topics we present to you a proportion definition and the concept of the constant of proportionality. After reading this text, you will be able to tell whether two parameters are directly proportional or inversely proportional.

Finally, we talk about some real-life proportion examples. You will see that proportional relationships are present everywhere in the world that surrounds us . Scientists use the law of multiple proportions while performing chemical reactions and accountants (or basically any taxpayer) should be familiar with the proportional tax definition to know how much money they have to pay to the state.

A proportion is a relationship between two quantities. It displays what portion of one part is contained in the whole. The result is typically seen as a fraction, but can also be represented with a colon, or as a decimal or percent. If your preference is in the percent-based display, we encourage you to visit our percentage calculator , it might also be a better choice for solving proportions with 100 100 100 as the denominator. It still follows the proportion definition but also enables you, for example, to calculate what percent of your birthday cake was eaten by your uncle 😊.

Just as a fraction is made up of two parts, a numerator and a denominator , the same holds true for a proportion. The upper number of proportion is the numerator, and the bottom part of the proportion is the denominator. To illustrate this, let's look at an example.

Suppose, that there is a cake cut into 12 slices. 5 of the 12 slices have been eaten (by that cheeky uncle). You want to know the proportion of slices remaining compared to the entire cake. You have 12 − 5 = 7 12-5 = 7 12 − 5 = 7 slices remaining, so the proportion is 7 / 12 7/12 7/12 . If you wish to rewrite this proportion using a colon, you could write it as 7 : 12 7:12 7 : 12 . Note that this proportion calculator will not represent the proportion in the latter form.

What is the process if we want to simplify or scale the proportion up to a larger, yet equivalent proportion? The next section on how to solve proportions explains this process.

The constant of proportionality connects ratios or products of two quantities. Sometimes it goes under the name of the coefficient of probability. We can write it in the form of an equation. If two variables are directly proportional , then the formula for the constant of proportionality looks like this:

where c c c is the constant of proportionality, while x x x and y y y are the two variables in a directly proportional relationship. If two variables are inversely proportional , then the coefficient of proportionality formula is:

where c c c is the constant of proportionality, while x x x and y y y are the two variables in an inversely proportional relationship.

It may sound like a very theoretical concept at first, but you will be surprised by the number of real-life applications of these constants. We use them so often that we don't realize that they are, mathematically speaking, constants of proportionality. Speed (or velocity) could be one of the most well-known examples. It is the ratio between distance and time. We can write it like this:

Where v v v is the velocity, s s s is the distance, and t t t stands for time.

Looks familiar, doesn't it? Yes, it is the same formula as for the constant of proportionality of two directly proportional variables . As this relationship is (by proportion definition) constant, then if we change one variable, the second one will also have to change. This way, by knowing our speed, we can also calculate the distance we will cover in 1 minute, 1 hour, 1 day, or any other period. With the value of time gradually increasing, the distance covered will also rise proportionally. The other way around, when the distance becomes shorter, then the time value will also have to decrease to maintain the constant ratio. Other popular examples of constants of proportionality of two directly proportional variables are concentration and density. In both of these examples, the directly proportional variables are mass and volume.

Now, we will talk about an example of the constant of proportionality of two inversely proportional variables. Let's stick with the theme of traveling. This time, let's have a look at the formula for the distance:

This equation seems to be more similar to the formula of the constant of proportionality of two inversely proportional variables , doesn't it? If the distance (constant) stays the same, then the change in velocity or time will cause a change of the other variable. As we are talking about an inversely proportional relationship, then an increase in velocity leads to a reduction in travel time. We observe this process in real life. A car traveling at a higher speed will reach its destination in a shorter period. In turn, if we want to prolong the travel time (e.g., to admire the countryside or to save on gas), then the velocity will have to decrease. The constant ratio has to be preserved. In the last paragraphs of this text, you will find some other real-life proportion examples and instructions on how to solve them.

Suppose that we have the same proportion of 7 / 12 7/12 7/12 but we want to scale it up to a larger, equivalent ratio with a denominator of 96 96 96 . To do this, we set up two equivalent proportions and solve for the missing part. The process of solving the proportion is as follows:

• Set up both fractions, labeling the missing part with any variable of your choice.
• Put an equal sign between the two ratios. Note that ratio and proportion are the same things.
• Isolate the variable by using cross multiplication, which is multiplying the denominator of one proportion by the numerator of the other and vice versa.
• Solve for the variable.
• Use the proportion calculator to check your answer. Now you know everything about solving proportions.

Solving the above example, we get:

We urge you to use the proportion calculator if you are solving proportions with larger numbers or those involving decimals.

The proportions can be used to adjust the amount of ingredients in a recipe for a larger number of guests. If you want to make pancakes, our pancake recipe calculator will tell you exactly what you have to prepare for a certain number of people.

There is a special ratio that occurs in nature and construction that is achieved when two quantities have the same proportion as the proportion of their sum to the larger of the two quantities. This is known as the golden ratio , which has a value of approximately 1.618 . We know that it sounds very complicated, but things will get clearer shortly. The formula for the golden ratio, given two quantities a a a and b b b is:

The golden ratio calculator is handy to compute this ratio.

The golden ratio can be seen in architecture and in general with shapes, such as the rectangle. A rectangle is golden if, with the given width w = a w = a w = a and length l = a + b l = a + b l = a + b , the ratio ( a + b ) / a = a / b (a+b)/a = a/b ( a + b ) / a = a / b yields the value of approximately 1.618. The golden ratio is the limit of the ratios of consecutive Fibonacci numbers and a non-terminating decimal. Thus we use the approximate value of 1.618. The golden rectangle calculator will compute the length and width that achieves the golden rectangle.

In geometry, the proportion calculator comes in handy when working with similar polygons. By definition, two polygons are said to be similar if their sides are in proportion. The most common application of this is with triangles. The law of sines for triangles is based on the fact that there is a proportional relationship between the sides and angles of a triangle.

The ability to solve proportions may also come in handy when dealing with chemical reactions. The law of multiple proportions is one of the most fundamental rules of stoichiometry, discovered by the British chemist John Dalton. It is the methodology of calculating the amounts of reactants and products of chemical reactions. For help with this, check out our theoretical yield calculator . Dalton's full statement looks like this:

If two elements form more than one compound between them, then the ratios of the masses of the second element which combine with a fixed mass of the first element will be ratios of small whole numbers.

Less scientifically speaking, if we divide masses of atoms of different elements forming a single particle (e.g., CO₂ – carbon dioxide), we always receive a constant ratio specific for this particle. The result will consist very often of small numbers.

Let's show it on an example and calculate the ratio for the sulfuric acid – H₂SO₄ :

• First, we have to count the numbers of atoms of particular elements forming the sulfuric acid. We have 2 atoms of hydrogen H, 1 atom of sulfur S, and 4 atoms of oxygen O.
• Secondly, we need to calculate the total mass of atoms of each element. Single hydrogen atom weighs 1 u (u stands for atomic mass unit), one atom of oxygen weighs 16 u, and one atom of sulfur has a mass of 32 u.

Joint mass of hydrogen atoms is: 2 × 1   u = 2   u 2 \times 1 \ \mathrm u = 2 \ \mathrm u 2 × 1   u = 2   u , The total mass of oxygen in the sulfuric acid particle is: 4 × 16   u = 64   u 4 \times 16 \ \mathrm u = 64 \ \mathrm u 4 × 16   u = 64   u , There is only one atom of sulfur in this particle, so the weight of sulfur is equal to 32   u 32 \ \mathrm u 32   u .

• The last thing to do is to divide all numbers one by another. The only condition is that we have to maintain all numbers as whole numbers. We cannot create fractions. It will look like this:
• And that’s it! We have received a ratio of small numbers for a particle of sulfuric acid.

Thanks to this knowledge we can tell that if we have 1 g (or any any other unit of mass) of hydrogen we need to add 16 g of sulfur and 32 g of oxygen to be sure that all atoms of each element will take part in the reaction and that there will not be any leftovers.

We have a task for you! Check (by using our proportions calculator) how many grams of hydrogen and oxygen can react fully with 352 grams of sulfur creating sulfuric acid (you can find the answer at the end of this text).

There are some other examples of proportional relationships in chemistry. The molarity shows the relationship between a number of moles of a certain solute (a dissolved substance) and a volume of solution. Some amateur chemists may be interested in the ABV (alcohol by volume), representing the relation between the volume of alcohol and the total volume of a drink.

After a quick visit to the world of chemistry, now it is time to see whether we can find our beloved proportions in finances. In fact, we can! Some of the taxes that we pay are proportional taxes. It means that the tax rate is fixed and everyone has to pay the same percentage of their income (or any other value undergoing proportional taxation).

Let’s illustrate it with an example. The proportional tax rate is 15%. Person X has an income of 15,000$ so the tax that he or she will have to pay at the end of the tax year will be equal to:

In turn, person Y earns 125,000$. However, the proportional tax definition requires the rate to be fixed, so to calculate the tax that he or she will have to pay we can use a similar formula:

Although the amount of tax paid by the person Y is greater than the tax of person X, it is the same part of their income as a whole, namely 15/100 or 15%.

Another type of tax (probably the most popular one) is a progressive tax. It is similar to the proportional tax as you also have to pay a certain percent of your income. The difference is that this percentage (or tax rate) changes with an increase of income. In progressive tax, there are specific tax rates for certain ranges of income value. For example, people of income under 100,000$ may have to pay 20% of their earnings, while people with an income over 100,000$ may have a tax rate of 25%.

Answer to the question: By the law of multiple proportions, 352 grams of sulfur can react fully with 22 grams of hydrogen and 704 grams of oxygen, creating sulfuric acid.

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2.6: Solve Proportions and Similar Figure Applications

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Learning Objectives

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

• Solve proportions
• Solve similar figure applications

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

• Solve \(\dfrac{n}{3}=30\). If you missed this problem, review Exercise 2.2.25 .
• The perimeter of a triangular window is 23 feet. The lengths of two sides are ten feet and six feet. How long is the third side? If you missed this problem, review Example 3.4.2

Solve Proportions

When two rational expressions are equal, the equation relating them is called a proportion .

Definition: PROPORTION

A proportion is an equation of the form \(\dfrac{a}{b}=\dfrac{c}{d}\), where \(b \ne 0\), \(d \ne 0\).

The proportion is read “a is to b, as c is to d"

The equation \(\dfrac{1}{2}=\dfrac{4}{8}\) is a proportion because the two fractions are equal.

The proportion \(\dfrac{1}{2}=\dfrac{4}{8}\) is read “1 is to 2 as 4 is to 8.”

Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.

Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let x be the number of teachers for 60 students and then set up the proportion:

\[\dfrac{1\,\text{teacher}}{20\,\text{students}}=\dfrac{x\,\text{teachers}}{60\,\text{students}}\nonumber\]

We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.

Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solve Rational Equations . We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.

Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.

Example \(\PageIndex{1}\)

\(\dfrac{x}{63}=\dfrac{4}{7}\).

Try It \(\PageIndex{1}\)

\(\dfrac{n}{84}=\dfrac{11}{12}\).

Try It \(\PageIndex{2}\)

\(\dfrac{y}{96}=\dfrac{13}{12}\).

Example \(\PageIndex{2}\)

\(\dfrac{144}{a}=\dfrac{9}{4}\).

Try It \(\PageIndex{3}\)

\(\dfrac{91}{b}=\dfrac{7}{5}\).

Try It \(\PageIndex{4}\)

\(\dfrac{39}{c}=\dfrac{13}{8}\).

Example \(\PageIndex{3}\)

\(\dfrac{n}{n+14}=\dfrac{5}{7}.\)

Try It \(\PageIndex{5}\)

\(\dfrac{y}{y+55}=\dfrac{3}{8}\).

Try It \(\PageIndex{6}\)

\(\dfrac{z}{z−84}=−\dfrac{1}{5}\).

Example \(\PageIndex{4}\)

\(\dfrac{p+12}{9}=\dfrac{p−12}{6}\).

Try It \(\PageIndex{7}\)

\(\dfrac{v+30}{8}=\dfrac{v+66}{12}\).

Try It \(\PageIndex{8}\)

\(\dfrac{2x+15}{9}=\dfrac{7x+3}{15}\).

To solve applications with proportions, we will follow our usual strategy for solving applications. But when we set up the proportion, we must make sure to have the units correct—the units in the numerators must match and the units in the denominators must match.

Example \(\PageIndex{5}\)

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

Try It \(\PageIndex{9}\)

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

Try It \(\PageIndex{10}\)

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

Example \(\PageIndex{6}\)

A 16-ounce iced caramel macchiato has 230 calories. How many calories are there in a 24-ounce iced caramel macchiato?

Try It \(\PageIndex{11}\)

At a fast-food restaurant, a 22-ounce chocolate shake has 850 calories. How many calories are in their 12-ounce chocolate shake? Round your answer to nearest whole number.

464 calories

Try It \(\PageIndex{12}\)

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

Example \(\PageIndex{7}\)

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

Try It \(\PageIndex{13}\)

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

590.4 Euros

Try It \(\PageIndex{14}\)

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

In the example above, we related the number of pesos to the number of dollars by using a proportion. We could say the number of pesos is proportional to the number of dollars. If two quantities are related by a proportion, we say that they are proportional.

Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures . If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.

Definition: SIMILAR FIGURES

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio.

For example, the two triangles in Figure are similar. Each side of ΔABC is 4 times the length of the corresponding side of ΔXYZ.

This is summed up in the Property of Similar Triangles.

Definition: PROPERTY OF SIMILAR TRIANGLES

• If ΔABC is similar to ΔXYZ

To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.

Definition: SOLVE GEOMETRY APPLICATIONS.

• Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
• Identify what we are looking for.
• Name what we are looking for by choosing a variable to represent it.
• Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
• Solve the equation using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Answer the question with a complete sentence.

Example \(\PageIndex{8}\)

ΔABC is similar to ΔXYZ. The lengths of two sides of each triangle are given in the figure.

Find the length of the sides of the similar triangles.

Try It \(\PageIndex{15}\)

ΔABC is similar to ΔXYZ. The lengths of two sides of each triangle are given in the figure.

Find the length of side a

Try It \(\PageIndex{16}\)

The next example shows how similar triangles are used with maps.

Example \(\PageIndex{9}\)

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. If the actual distance from Los Angeles to Las Vegas is 270 miles find the distance from Los Angeles to San Francisco.

Try It \(\PageIndex{17}\)

On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

Try It \(\PageIndex{18}\)

Using the map above, find the distance from Portland to Boise.

We can use similar figures to find heights that we cannot directly measure.

Example \(\PageIndex{10}\)

Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.

Try It \(\PageIndex{19}\)

A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

Try It \(\PageIndex{20}\)

A pine tree casts a shadow of 80 feet next to a 30-foot tall building which casts a 40 feet shadow. How tall is the pine tree?

Key Concepts

• Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.