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Solve Proportions and their Applications
Use the definition of proportion
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.
The equation \(\frac{1}{2}=\frac{4}{8}\) is a proportion because the two fractions are equal. The proportion \(\frac{1}{2}=\frac{4}{8}\) is read \(\text{“}1\) is to \(2\) as \(4\) is to \(8\text{”.}\)
If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion \(\frac{\text{20 students}}{\text{1 teacher}}=\frac{\text{60 students}}{\text{3 teachers}}\) we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.
Example
Try it.
Write each sentence as a proportion:
- ⓐ \(\ 3\) is to \(7\) as \(15\) is to \(35.\)
- ⓑ \(\ 5\) hits in \(8\) at bats is the same as \(30\) hits in \(48\) at-bats.
- ⓒ \(\ \text{\$1.50}\) for \(6\) ounces is equivalent to \(\text{\$2.25}\) for \(9\) ounces.
Solution
| ⓐ | |
| 3 is to 7 as 15 is to 35. | |
| Write as a proportion. | \(\frac{3}{7}=\frac{15}{35}\) |
| ⓑ | |
| 5 hits in 8 at-bats is the same as 30 hits in 48 at-bats. | |
| Write each fraction to compare hits to at-bats. | \(\frac{\text{hits}}{\text{at-bats}}=\frac{\text{hits}}{\text{at-bats}}\) |
| Write as a proportion. | \(\frac{5}{8}=\frac{30}{48}\) |
| ⓒ | |
| $1.50 for 6 ounces is equivalent to $2.25 for 9 ounces. | |
| Write each fraction to compare dollars to ounces. | \(\frac{\$}{\text{ounces}}=\frac{\$}{\text{ounces}}\) |
| Write as a proportion. | \(\frac{1.50}{6}=\frac{2.25}{9}\) |
Look at the proportions \(\frac{1}{2}=\frac{4}{8}\) and \(\frac{2}{3}=\frac{6}{9}.\) 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?
Condensed — the full section is in OpenStax Prealgebra 2e.
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
Try it.
Solve: \(\frac{x}{63}=\frac{4}{7}.\)
Solution
| To isolate \(x\), multiply both sides by the LCD, 63. | ||
| Simplify. | ||
| Divide the common factors. | ||
| Check: To check our answer, we substitute into the original proportion. | ||
| Show common factors. | ||
| Simplify. |
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
Try it.
Solve: \(\frac{144}{a}=\frac{9}{4}.\)
Solution
Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.
| Find the cross products and set them equal. | ||
| Simplify. | ||
| Divide both sides by 9. | ||
| Simplify. | ||
| Check your answer. | ||
| Show common factors.. | ||
| Simplify. |
Another method to solve this would be to multiply both sides by the LCD, \(4a.\) Try it and verify that you get the same solution.
Example
Try it.
Solve: \(\frac{52}{91}=\frac{-4}{y}.\)
Solution
| Find the cross products and set them equal. | ||
| Simplify. | ||
| Divide both sides by 52. | ||
| Simplify. | ||
| Check: | ||
| Show common factors. | ||
| Simplify. |
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
Try it.
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?
Solution
| Identify what you are asked to find. | How many ml of acetaminophen the doctor will prescribe |
| Choose a variable to represent it. | Let \(a=\) ml of acetaminophen. |
| Write a sentence that gives the information to find it. | If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds? |
| Translate into a proportion. | |
| Substitute given values—be careful of the units. | |
| Multiply both sides by 80. | |
| Multiply and show common factors. | |
| Simplify. | |
| Check if the answer is reasonable. | |
| Yes. Since 80 is about 3 times 25, the medicine should be about 3 times 5. | |
| Write a complete sentence. | The pediatrician would prescribe 16 ml of acetaminophen to Zoe. |
You could also solve this proportion by setting the cross products equal.
Example
Try it.
One brand of microwave popcorn has \(120\) calories per serving. A whole bag of this popcorn has \(3.5\) servings. How many calories are in a whole bag of this microwave popcorn?
Solution
| Identify what you are asked to find. | How many calories are in a whole bag of microwave popcorn? |
| Choose a variable to represent it. | Let \(c=\) number of calories. |
| Write a sentence that gives the information to find it. | If there are 120 calories per serving, how many calories are in a whole bag with 3.5 servings? |
| Translate into a proportion. | |
| Substitute given values. | |
| Multiply both sides by 3.5. | |
| Multiply. | |
| Check if the answer is reasonable. | |
| Yes. Since 3.5 is between 3 and 4, the total calories should be between 360 (3⋅120) and 480 (4⋅120). | |
| Write a complete sentence. | The whole bag of microwave popcorn has 420 calories. |
Condensed — the full section is in OpenStax Prealgebra 2e.
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, \(\text{60\%}=\frac{60}{100}\) and we can simplify \(\frac{60}{100}=\frac{3}{5}.\) Since the equation \(\frac{60}{100}=\frac{3}{5}\) shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:
\[\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}\]\[\ \frac{3}{5}=\frac{60}{100}\]If we restate the problem in the words of a proportion, it may be easier to set up the proportion:
\[\text{The amount is to the base as the percent is to one hundred.}\]We could also say:
\[\text{The amount out of the base is the same as the percent out of one hundred.}\]First we will practice translating into a percent proportion. Later, we’ll solve the proportion.
Example
Try it.
Translate to a proportion. What number is \(\text{75\%}\) of \(90?\)
Solution
If you look for the word "of", it may help you identify the base.
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | |
| Set up the proportion. Let \(n=\text{number}\). | \(\frac{n}{90}=\frac{75}{100}\) |
Example
Try it.
Translate to a proportion. \(19\) is \(\text{25\%}\) of what number?
Solution
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | |
| Set up the proportion. Let \(n=\text{number}\). | \(\frac{19}{n}=\frac{25}{100}\) |
Example
Try it.
Translate to a proportion. What percent of \(27\) is \(9?\)
Solution
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | |
| Set up the proportion. Let \(p=\text{percent}\). | \(\frac{9}{27}=\frac{p}{100}\) |
Translate and Solve Percent Proportions
Now that we have written percent equations as proportions, we are ready to solve the equations.
Example
Try it.
Translate and solve using proportions: What number is \(\text{45\%}\) of \(80?\)
Solution
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | |
| Set up the proportion. Let \(n=\) number. | |
| Find the cross products and set them equal. | |
| Simplify. | |
| Divide both sides by 100. | |
| Simplify. | |
| Check if the answer is reasonable. | |
| Yes. 45 is a little less than half of 100 and 36 is a little less than half 80. | |
| Write a complete sentence that answers the question. | 36 is 45% of 80. |
In the next example, the percent is more than \(100,\) which is more than one whole. So the unknown number will be more than the base.
Example
Try it.
Translate and solve using proportions: \(\text{125\%}\) of \(25\) is what number?
Solution
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | |
| Set up the proportion. Let \(n=\) number. | |
| Find the cross products and set them equal. | |
| Simplify. | |
| Divide both sides by 100. | |
| Simplify. | |
| Check if the answer is reasonable. | |
| Yes. 125 is more than 100 and 31.25 is more than 25. | |
| Write a complete sentence that answers the question. | 125% of 25 is 31.25. |
Percents with decimals and money are also used in proportions.
Example
Try it.
Translate and solve: \(\text{6.5\%}\) of what number is \(\text{\$1.56}?\)
Solution
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | |
| Set up the proportion. Let\(n=\) number. | |
| Find the cross products and set them equal. | |
| Simplify. | |
| Divide both sides by 6.5 to isolate the variable. | |
| Simplify. | |
| Check if the answer is reasonable. | |
| Yes. 6.5% is a small amount and $1.56 is much less than $24. | |
| Write a complete sentence that answers the question. | 6.5% of $24 is $1.56. |
Condensed — the full section is in OpenStax Prealgebra 2e.
Solve Proportions and their Applications
In the following exercises, convert each percent to ⓐ a decimal ⓑ a simplified fraction.
In the following exercises, convert each fraction to a percent. (Round to \(3\) decimal places if needed.)
In the following exercises, solve the percent problem.
Practice (40)
Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.
-
Simplify: \(\frac{\frac{1}{3}}{4}.\)
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\(\frac{1}{12}\)
-
Solve: \(\frac{x}{4}=20.\)
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\(80\)
-
Write as a rate: Sale rode his bike \(24\) miles in \(2\) hours.
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\(\frac{24\ \text{miles}}{2\ \text{hours}}\)
-
Write each sentence as a proportion:
- ⓐ \(\ 3\) is to \(7\) as \(15\) is to \(35.\)
- ⓑ \(\ 5\) hits in \(8\) at bats is the same as \(30\) hits in \(48\) at-bats.
- ⓒ \(\ \text{\$1.50}\) for \(6\) ounces is equivalent to \(\text{\$2.25}\) for \(9\) ounces.
Mutasd meg a választ!
ⓐ 3 is to 7 as 15 is to 35. Write as a proportion. \(\frac{3}{7}=\frac{15}{35}\) ⓑ 5 hits in 8 at-bats is the same as 30 hits in 48 at-bats. Write each fraction to compare hits to at-bats. \(\frac{\text{hits}}{\text{at-bats}}=\frac{\text{hits}}{\text{at-bats}}\) Write as a proportion. \(\frac{5}{8}=\frac{30}{48}\) ⓒ $1.50 for 6 ounces is equivalent to $2.25 for 9 ounces. Write each fraction to compare dollars to ounces. \(\frac{\$}{\text{ounces}}=\frac{\$}{\text{ounces}}\) Write as a proportion. \(\frac{1.50}{6}=\frac{2.25}{9}\) -
Write each sentence as a proportion:
- ⓐ \(\ 5\) is to \(9\) as \(20\) is to \(36.\)
- ⓑ \(\ 7\) hits in \(11\) at-bats is the same as \(28\) hits in \(44\) at-bats.
- ⓒ \(\ \text{\$2.50}\) for \(8\) ounces is equivalent to \(\text{\$3.75}\) for \(12\) ounces.
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- ⓐ \(\ \frac{5}{9}=\frac{20}{36}\\)
- ⓑ \(\ \frac{7}{11}=\frac{28}{44}\\)
- ⓒ \(\frac{2.50}{8}=\frac{3.75}{12}\\)
-
Write each sentence as a proportion:
- ⓐ \(\ 6\) is to \(7\) as \(36\) is to \(42.\)
- ⓑ \(\ 8\) adults for \(36\) children is the same as \(12\) adults for \(54\) children.
- ⓒ \(\ \text{\$3.75}\) for \(6\) ounces is equivalent to \(\text{\$2.50}\) for \(4\) ounces.
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- ⓐ \(\ \frac{6}{7}=\frac{36}{42}\)
- ⓑ \(\ \frac{8}{36}=\frac{12}{54}\)
- ⓒ \(\frac{3.75}{6}=\frac{2.50}{4}\)
-
Determine whether each equation is a proportion:
- ⓐ \(\ \frac{4}{9}=\frac{12}{28}\)
- ⓑ \(\frac{17.5}{37.5}=\frac{7}{15}\)
Mutasd meg a választ!
To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.
ⓐ Find the cross products. \(28⋅4=112\ 9⋅12=108\) Since the cross products are not equal, \(28\cdot 4\ne 9\cdot 12,\) the equation is not a proportion.
ⓑ Find the cross products. \(15⋅17.5=262.5\ 37.5⋅7=262.5\)
Since the cross products are equal, \(15\cdot 17.5=37.5\cdot 7,\) the equation is a proportion.
-
Determine whether each equation is a proportion:
- ⓐ \(\ \frac{7}{9}=\frac{54}{72}\)
- ⓑ \(\frac{24.5}{45.5}=\frac{7}{13}\)
Mutasd meg a választ!
- ⓐ no
- ⓑ yes
-
Determine whether each equation is a proportion:
- ⓐ \(\ \frac{8}{9}=\frac{56}{73}\\)
- ⓑ \(\ \frac{28.5}{52.5}=\frac{8}{15}\)
Mutasd meg a választ!
- ⓐ no
- ⓑ no
-
Solve: \(\frac{x}{63}=\frac{4}{7}.\)
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To isolate \(x\), multiply both sides by the LCD, 63. Simplify. Divide the common factors. Check: To check our answer, we substitute into the original proportion. Show common factors. Simplify. -
Solve the proportion: \(\frac{n}{84}=\frac{11}{12}.\)
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77
-
Solve the proportion: \(\frac{y}{96}=\frac{13}{12}.\)
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104
-
Solve: \(\frac{144}{a}=\frac{9}{4}.\)
Mutasd meg a választ!
Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.
Find the cross products and set them equal. Simplify. Divide both sides by 9. Simplify. Check your answer. Show common factors.. Simplify. Another method to solve this would be to multiply both sides by the LCD, \(4a.\) Try it and verify that you get the same solution.
-
Solve the proportion: \(\frac{91}{b}=\frac{7}{5}.\)
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65
-
Solve the proportion: \(\frac{39}{c}=\frac{13}{8}.\)
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24
-
Solve: \(\frac{52}{91}=\frac{-4}{y}.\)
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Find the cross products and set them equal. Simplify. Divide both sides by 52. Simplify. Check: Show common factors. Simplify. -
Solve the proportion: \(\frac{84}{98}=\frac{-6}{x}.\)
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−7
-
Solve the proportion: \(\frac{-7}{y}=\frac{105}{135}.\)
Mutasd meg a választ!
−9
-
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?
Mutasd meg a választ!
Identify what you are asked to find. How many ml of acetaminophen the doctor will prescribe Choose a variable to represent it. Let \(a=\) ml of acetaminophen. Write a sentence that gives the information to find it. If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds? Translate into a proportion. Substitute given values—be careful of the units. Multiply both sides by 80. Multiply and show common factors. Simplify. Check if the answer is reasonable. Yes. Since 80 is about 3 times 25, the medicine should be about 3 times 5. Write a complete sentence. The pediatrician would prescribe 16 ml of acetaminophen to Zoe. You could also solve this proportion by setting the cross products equal.
-
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?
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12 ml
-
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?
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180 mg
-
One brand of microwave popcorn has \(120\) calories per serving. A whole bag of this popcorn has \(3.5\) servings. How many calories are in a whole bag of this microwave popcorn?
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Identify what you are asked to find. How many calories are in a whole bag of microwave popcorn? Choose a variable to represent it. Let \(c=\) number of calories. Write a sentence that gives the information to find it. If there are 120 calories per serving, how many calories are in a whole bag with 3.5 servings? Translate into a proportion. Substitute given values. Multiply both sides by 3.5. Multiply. Check if the answer is reasonable. Yes. Since 3.5 is between 3 and 4, the total calories should be between 360 (3⋅120) and 480 (4⋅120). Write a complete sentence. The whole bag of microwave popcorn has 420 calories. -
Marissa loves the Caramel Macchiato at the coffee shop. The \(16\) oz. medium size has \(240\) calories. How many calories will she get if she drinks the large \(20\) oz. size?
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300
-
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?
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5 pieces
-
Josiah went to Mexico for spring break and changed \(\text{\$325}\) dollars into Mexican pesos. At that time, the exchange rate had \(\text{\$1}\) U.S. is equal to \(12.54\) Mexican pesos. How many Mexican pesos did he get for his trip?
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Identify what you are asked to find. How many Mexican pesos did Josiah get? Choose a variable to represent it. Let \(p=\) number of pesos. Write a sentence that gives the information to find it. If $1 U.S. is equal to 12.54 Mexican pesos, then $325 is how many pesos? Translate into a proportion. Substitute given values. The variable is in the denominator, so find the cross products and set them equal. Simplify. Check if the answer is reasonable. Yes, $100 would be $1,254 pesos. $325 is a little more than 3 times this amount. Write a complete sentence. Josiah has 4075.5 pesos for his spring break trip. -
Yurianna is going to Europe and wants to change \(\text{\$800}\) dollars into Euros. At the current exchange rate, \(\text{\$1}\) US is equal to \(0.738\) Euro. How many Euros will she have for her trip?
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590 Euros
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Corey and Nicole are traveling to Japan and need to exchange \(\text{\$600}\) into Japanese yen. If each dollar is \(94.1\) yen, how many yen will they get?
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56,460 yen
-
Translate to a proportion. What number is \(\text{75\%}\) of \(90?\)
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If you look for the word "of", it may help you identify the base.
Identify the parts of the percent proportion. Restate as a proportion. Set up the proportion. Let \(n=\text{number}\). \(\frac{n}{90}=\frac{75}{100}\) -
Translate to a proportion: What number is \(\text{60\%}\) of \(105?\)
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\(\frac{n}{105}=\frac{60}{100}\)
-
Translate to a proportion: What number is \(\text{40\%}\) of \(85?\)
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\(\frac{n}{85}=\frac{40}{100}\)
-
Translate to a proportion. \(19\) is \(\text{25\%}\) of what number?
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Identify the parts of the percent proportion. Restate as a proportion. Set up the proportion. Let \(n=\text{number}\). \(\frac{19}{n}=\frac{25}{100}\) -
Translate to a proportion: \(36\) is \(\text{25\%}\) of what number?
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\(\frac{36}{n}=\frac{25}{100}\)
-
Translate to a proportion: \(27\) is \(\text{36\%}\) of what number?
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\(\frac{27}{n}=\frac{36}{100}\)
-
Translate to a proportion. What percent of \(27\) is \(9?\)
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Identify the parts of the percent proportion. Restate as a proportion. Set up the proportion. Let \(p=\text{percent}\). \(\frac{9}{27}=\frac{p}{100}\) -
Translate to a proportion: What percent of \(52\) is \(39?\)
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\(\frac{n}{100}=\frac{39}{52}\)
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Translate to a proportion: What percent of \(92\) is \(23?\)
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\(\frac{n}{100}=\frac{23}{92}\)
-
Translate and solve using proportions: What number is \(\text{45\%}\) of \(80?\)
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Identify the parts of the percent proportion. Restate as a proportion. Set up the proportion. Let \(n=\) number. Find the cross products and set them equal. Simplify. Divide both sides by 100. Simplify. Check if the answer is reasonable. Yes. 45 is a little less than half of 100 and 36 is a little less than half 80. Write a complete sentence that answers the question. 36 is 45% of 80. -
Translate and solve using proportions: What number is \(\text{65\%}\) of \(40?\)
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\(\frac{n}{40}=\frac{65}{100};\ n=26\)
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Translate and solve using proportions: What number is \(\text{85\%}\) of \(40?\)
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\(\frac{n}{40}=\frac{85}{100};\ n=34\)
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Translate and solve using proportions: \(\text{125\%}\) of \(25\) is what number?
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Identify the parts of the percent proportion. Restate as a proportion. Set up the proportion. Let \(n=\) number. Find the cross products and set them equal. Simplify. Divide both sides by 100. Simplify. Check if the answer is reasonable. Yes. 125 is more than 100 and 31.25 is more than 25. Write a complete sentence that answers the question. 125% of 25 is 31.25.
Symbols used here
The two sides are different.
Both signs at once: x = 3 ± 2 means 5 and 1.
Inequalities that allow equality; < and > exclude it.
Equal to the precision shown, not exactly.
The non-negative number whose square (n-th power) is x.
What is left after dividing a by n.
Per hundred: 15% = 15/100.
a for every b; a divided by b.
How to: Solve Proportions and their Applications
- Use the definition of proportion
- Solve proportions
- Solve applications using proportions
- Write percent equations as proportions
- Translate and solve percent proportions
- ⓐ no
- ⓑ yes
- ⓐ no
Questions people ask
Why does the order of operations matter?
Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.
How do I check an arithmetic answer?
Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.
Why are fractions harder than decimals?
They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.
Próbáld a sajátodat.
Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.