8.7 Solve Proportion and Similar Figure Applications

Learning objectives

Learning objectives

• Solve proportions

• Solve similar figure applications

Solve Proportions

When two rational expressions 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 “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:

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.

So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.

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

Example 8.71

Solution

When we work with proportions, we exclude values that would make either denominator zero, just like we do for all rational expressions. What value(s) should be excluded for the proportion in the next example?

Example 8.72

Solution

Example 8.73

Solution

Example 8.74

Solution

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 8.75

Solution

Example 8.76

Solution

Example 8.77

Solution

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.

For example, the two triangles in Figure 8.2 are similar. Each side of $\Delta{}ABC$ is 4 times the length of the corresponding side of $\Delta{}XYZ$.

This is summed up in the Property of Similar Triangles.

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

Example 8.78

Solution

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

Example 8.79

Solution

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

Example 8.80

Solution

Section 8.7 Exercises

Section 8.7 Exercises

Solve Proportions

In the following exercises, solve.

Solve Similar Figure Applications

In the following exercises, $\Delta{}ABC$ is similar to $\Delta{}XYZ$. Find the length of the indicated side.

In the following exercises, $\Delta{}DEF$ is similar to $\Delta{}NPQ$.

In the following two exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle whose sides are shown in the figure below. The actual distance from New York to Chicago is 800 miles.

In the following two exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle whose sides are shown in the figure below. The actual distance from Atlanta to New Orleans is 420 miles.

Everyday Math

Writing Exercises

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

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Solve the proportion: $\frac{n}{84}=\frac{11}{12}.$

2. Solve the proportion: $\frac{y}{96}=\frac{13}{12}.$

3. Solve the proportion: $\frac{91}{b}=\frac{7}{5}.$

4. Solve the proportion: $\frac{39}{c}=\frac{13}{8}.$

1. Solve the proportion: $\frac{y}{y+55}=\frac{3}{8}.$

2. Solve the proportion: $\frac{z}{z-84}=-\frac{1}{5}.$

3. Solve: $\frac{v+30}{8}=\frac{v+66}{12}.$

4. Solve: $\frac{2x+15}{9}=\frac{7x+3}{15}.$

1. 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?

2. 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?

3. 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.

4. 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?

1. 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?

2. 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?

3. 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.

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

1. 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?

2. 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?

3. $\frac{x}{56}=\frac{7}{8}$

4. $\frac{49}{63}=\frac{z}{9}$

1. $\frac{5}{a}=\frac{65}{117}$

2. $\frac{98}{154}=\frac{-7}{p}$

3. $\frac{a}{-8}=\frac{-42}{48}$

4. $\frac{2.7}{j}=\frac{0.9}{0.2}$

1. $\frac{a}{a+12}=\frac{4}{7}$

2. $\frac{c}{c-104}=-\frac{5}{8}$

3. $\frac{m+90}{25}=\frac{m+30}{15}$

4. $\frac{2p+4}{8}=\frac{p+18}{6}$

1. 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 Jocelyn, who weighs 45 pounds?

2. A veterinarian prescribed Sunny, a 65 pound dog, an antibacterial medicine in case an infection emerges after her teeth were cleaned. If the dosage is 5 mg for every pound, how much medicine was Sunny given?

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

4. A new 7 ounce lemon ice drink is advertised for having only 140 calories. How many ounces could Sally drink if she wanted to drink just 100 calories?

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

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

3. When traveling to Great Britain, Bethany exchanged her $900 into 570 British pounds. How many pounds did she receive for each American dollar?

4. Ronald needs a morning breakfast drink that will give him at least 390 calories. Orange juice has 130 calories in one cup. How many cups does he need to drink to reach his calorie goal?

1. Elizabeth is returning to the United States from Canada. She changes the remaining 300 Canadian dollars she has to $230.05 in American dollars. What was$1 worth in Canadian dollars?

2. A golden retriever weighing 85 pounds has diarrhea. His medicine is prescribed as 1 teaspoon per 5 pounds. How much medicine should he be given?

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

4. side b

1. Find the length of side d.

2. Find the actual distance from New York to Memphis.

3. Find the actual distance from New Orleans to Miami.

4. A 2 foot tall dog casts a 3 foot shadow at the same time a cat casts a one foot shadow. How tall is the cat?

1. The tower portion of a windmill is 212 feet tall. A six foot tall person standing next to the tower casts a seven foot shadow. How long is the windmill’s shadow?

2. Heart Rate At the gym, Carol takes her pulse for 10 seconds and counts 19 beats.

1. ⓐ How many beats per minute is this?

2. ⓑ Has Carol met her target heart rate of 140 beats per minute?

3. Cost of a Road Trip Jesse’s car gets 30 miles per gallon of gas.

1. ⓐ If Las Vegas is 285 miles away, how many gallons of gas are needed to get there and then home?

2. ⓑ If gas is $3.09 per gallon, what is the total cost of the gas for the trip?

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

1. Cooking Natalia’s pasta recipe calls for 2 pounds of pasta for 1 quart of sauce. How many pounds of pasta should Natalia cook if she has 2.5 quarts of sauce?

2. Marisol solves the proportion $\frac{144}{a}=\frac{9}{4}$ by ‘cross multiplying’, so her first step looks like $4\cdot{}144=9\cdot{}a.$ Explain how this differs from the method of solution shown in Example 8.72.