4.3: Write and Solve Proportions by Using Equivalent Rates
Have you ever faced a reading challenge? Take a look at this dilemma.
Jamie looked at the clock. She still had 20 minutes to read in the reading challenge.
Jamie looked up at the clock when the bell rang. He had read 15 pages in 20 minutes. If she had read 15 pages in 20 minutes, how many pages would she read in 40 minutes?
Do you know how to figure this out? You can use equivalent rates to help you solve this problem. Pay attention to this Concept and you will understand the solution by the end of it.
Guidance
A ratio is a comparison between two quantities or numbers. Ratios can be written in fraction form, with a colon or by using the word “to”.
Sometimes, you will compare ratios. Sometimes one ratio will be greater than another, and other times they can be equal or equivalent. When you have two equal ratios, you have a proportion.
A proportion is created when two ratios are equal, or we can say that two equal ratios form a proportion.
We can write a proportion when we know that two ratios are equivalent.
\begin{align*}1 : 2 = 2 : 4\end{align*}
These two ratios are equivalent. We can say that the two ratios form a proportion.
Do these two ratios form a proportion?
\begin{align*}\frac{3}{4}\end{align*}
To figure this out, we have to figure out if the two ratios are equivalent. If they are, then we know that they form a proportion. If not, then they don’t. To figure this out, we can simplify the ratios.
\begin{align*}& \qquad \ \frac{3}{4} \ is \ in \ simplest \ form.\\
& 4 : 24 \ can \ be \ written \ as \ \frac{4}{24} = \frac{1}{6}\\
& \qquad \qquad \quad \ \ \frac{3}{4} \ne \frac{1}{6}\end{align*}
These ratios do not form a proportion.
These proportions were given to you. You can also write your own proportions.
To write a proportion, set two equivalent fractions equal to each other, using the information in the problem.
If you know the ratio of girls to boys in a class is 2 : 3, and you know there are 24 boys in the class, you can write a proportion in order to find the number of girls in the class.
The most important thing to remember when writing a proportion is to keep the units the same in both ratios.
\begin{align*}\frac{girls}{boys}: \frac{2}{3} = \frac{x}{24}\end{align*}
You know the fractions are equivalent because each shows the ratio of girls to boys in the class. The first fraction shows the known ratio of girls to boys. The second ratio shows the known number of boys in the class, 24, and uses a variable to stand for the unknown number of girls.
Now let's use equivalent rates to solve a proportion.
The ratio of teachers to students in a certain school is 2 : 25. If there are 400 students in the eighthgrade class, how many teachers are there?
First set up a proportion. The problem gives a ratio of teachers to students, so set up two equivalent ratios comparing teachers to students.
\begin{align*}\frac{teachers}{students} = \frac{8^{th} \ grade \ teachers}{8^{th} \ grade \ students}\end{align*}
You can see that we are comparing teachers to students in both ratios. The first one shows the ration in the whole school and the second ratio represents the eighth grade ratios. Next, we fill in the given information.
\begin{align*}\frac{2}{25} = \frac{x}{400}\end{align*}
Now use what you know about equivalent ratios to solve the proportion.
Look at the denominators. You know that the first fraction, when the numerator and denominator are multiplied by some number, will equal the second fraction. What number, when multiplied by 25, will equal 400? Since \begin{align*}25 \times 16 = 400\end{align*}
\begin{align*}2 \times 16 = 32\end{align*}
There are 32 teachers in the eighthgrade class.
Note: You can check that your answer is correct by making sure that the two ratios are equivalent.
\begin{align*}\frac{32}{400} = \frac{8}{100} = \frac{2}{25}\end{align*}
Since the second ratio simplifies to the first, the ratios are equivalent.
Yes it is. Just remember that what you do to the numerator you have to do the denominator. If you can remember to always apply this rule, then you will create equal ratios.
Solve each proportion by using equal ratios.
Example A
\begin{align*}\frac{3}{4} = \frac{6}{x}\end{align*}
Solution: \begin{align*}x = 8\end{align*}
Example B
\begin{align*}\frac{9}{50} = \frac{x}{100}\end{align*}
Solution: \begin{align*}x = 18\end{align*}
Example C
\begin{align*}\frac{3.5}{7} = \frac{x}{35}\end{align*}
Solution: \begin{align*}x = 17.5\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
Jamie read 15 pages in 20 minutes. She wonders how many pages she will read in 40 minutes.
We can set up a proportion and look for an equivalent rate.
\begin{align*}\frac{pages}{minutes} = \frac{15}{20} = \frac{x}{40}\end{align*}
Here is our proportion.
Next, we can look at the relationship between the denominators.
\begin{align*}20 \times 2 = 40\end{align*}
What we do to the bottom, we can do to the top. This will give us the equivalent rate.
\begin{align*}15 \times 2 = 30\end{align*}
At this rate, Jamie will read 30 pages in 40 minutes.
Vocabulary
 Ratio
 a comparison between two quantities. Ratios can be written in fraction form, with a colon or by using the word “to”.
 Equivalent
 means equal.
 Proportion
 formed when two ratios are equivalent. We compare two ratios, they are equal and so they form a proportion.
Guided Practice
Here is one for you to try on your own.
Write a proportion to describe this situation.
The proportion of red paper to white paper in a stack is 2 to 7. If there are 32 red pieces of paper, what proportion could be used to find the number of pieces of white paper?
Solution
Write the known ratio of red paper to white paper as the first fraction: \begin{align*}\frac{2}{7}\end{align*}
Now write the second ratio, using \begin{align*}x\end{align*}
\begin{align*}\frac{\text{red paper}}{\text{white paper}} : \frac{2}{7} = \frac{32}{x}\end{align*}
The proportion \begin{align*}\frac{2}{7} = \frac{32}{x}\end{align*}
Video Review
Practice
Directions: Solve each proportion using equal ratios.

\begin{align*}\frac{3}{4} = \frac{x}{12}\end{align*}
34=x12 
\begin{align*}\frac{5}{6} = \frac{x}{12}\end{align*}
56=x12 
\begin{align*}\frac{4}{7} = \frac{8}{y}\end{align*}
47=8y 
\begin{align*}\frac{2}{3} = \frac{12}{y}\end{align*}
23=12y 
\begin{align*}\frac{4}{5} = \frac{44}{y}\end{align*}
45=44y 
\begin{align*}\frac{12}{13} = \frac{x}{26}\end{align*}
1213=x26 
\begin{align*}\frac{9}{10} = \frac{81}{y}\end{align*}
910=81y 
\begin{align*}\frac{6}{7} = \frac{18}{y}\end{align*}
67=18y 
\begin{align*}\frac{7}{8} = \frac{x}{56}\end{align*}
78=x56 
\begin{align*}\frac{12}{14} = \frac{36}{x}\end{align*}
1214=36x 
\begin{align*}\frac{6}{4} = \frac{x}{12}\end{align*}
64=x12 
\begin{align*}\frac{12}{14} = \frac{24}{x}\end{align*}
1214=24x 
\begin{align*}\frac{13}{14} = \frac{x}{42}\end{align*}
1314=x42 
\begin{align*}\frac{1.5}{4} = \frac{x}{8}\end{align*}
1.54=x8 
\begin{align*}\frac{3.5}{4.5} = \frac{x}{9}\end{align*}
3.54.5=x9 
\begin{align*}\frac{9}{14} = \frac{108}{x}\end{align*}
914=108x
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Here you'll write and solve proportions by using equivalent rates.