8.3: Solving Proportions
Introduction
Stocking Shelves
Chase, Marc and Kris are all working at the supermarket in the stock room. After school, they work to stock the shelves at the local supermarket with all kinds of cans. In fact, they have gotten quite fast at it and they love to have contests to see who is the fastest. Often the one who loses has to treat the others to ice cream after work.
It takes Chase 15 minutes to stock three shelves of canned goods, and it takes Marc 45 minutes to stock nine shelves.
“I am definitely faster,” Chase tells Marc one afternoon.
“I don’t think so, I think we are both the same.” Marc disagrees.
Chase and Marc continue to argue. Who is correct?
Their friend Kris stocks shelves at the same rate as Chase does. If he stocks 12 shelves at this rate, how many minutes does it take him?
This lesson is all about proportions and you will need proportions to help these boys sort out their argument.
What You Will Learn
By the end of this lesson you will be able to demonstrate the following skills:
- Use cross products to identify proportions.
- Solve for the unknown part of given proportions using mental math.
- Solve for the unknown part of given proportions using related equations.
- Solve real-world problems involving proportions.
Teaching Time
Proportions are everywhere in the world around us. Proportions are comparisons that we make between different things. You will often hear the words “in proportion” meaning that there is a relationship between things. What is the relationship of a proportion? That is exactly what this lesson is going to work on.
What is a proportion?
A proportion is two equal ratios. Remember that a ratio compares two quantities; well, a proportion compares two equal ratios.
While ratios can be written in three different ways, often you will see proportions written as equal fractions.
Let’s look at a proportion.
Example
\begin{align*}\frac{4}{12} = \frac{1}{3}\end{align*}
Here we have two ratios. We have four compared to twelve and one compared to three. These two ratios form a proportion. Simplified, they equal the same thing. You can simplify four-twelfths and it equals one-third.
Sometimes one of the trickiest things is figuring out if two ratios form a proportion. In the example above, we can see the equals sign letting us know that the ratios form a proportion.
How can we tell if two ratios form a proportion?
There are two different ways to figure this out. The first has already been mentioned and that is to simplify the two ratios and see if they are equal.
Example
\begin{align*}\frac{1}{4}\end{align*} and \begin{align*}\frac{5}{20}\end{align*}
One-fourth is already in simplest form, we leave that one alone. If we simplify five-twentieths, we get one-fourth as an answer. One-fourth is equal to one-fourth, so these two ratios do form a proportion.
Example
\begin{align*}\frac{2}{8}\end{align*} and \begin{align*}\frac{3}{6}\end{align*}
If we simplify these two fractions we get two different answers. Two-eighths simplifies to one-fourth. Three-sixths simplifies to one-half. The two ratios are not equal. Therefore, they DO NOT form a proportion.
The second way of figuring out if two ratios form a proportion is to cross multiply or to use cross products.
What is a cross product?
A cross product is when you multiply the numerator of one ratio with the denominator of another. Essentially you multiply on the diagonals. If the product is the same, then the two ratios form a proportion.
Example
\begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{4}{6}\end{align*}
Let’s use cross products here.
\begin{align*}2 \times 6 &= 12\\ 3 \times 4 &= 12\\ 12 &= 12\end{align*}
The two ratios form a proportion.
We can use cross products to figure out whether or not two ratios form a proportion.
Try a few of these on your own. Use cross products to determine if the two ratios form a proportion. Write yes if they form a proportion and no if they do not.
- \begin{align*}\frac{2}{5}\end{align*} and \begin{align*}\frac{5}{9}\end{align*}
- \begin{align*}\frac{3}{6}\end{align*} and \begin{align*}\frac{5}{10}\end{align*}
- \begin{align*}\frac{4}{7}\end{align*} and \begin{align*}\frac{12}{28}\end{align*}
Take a few minutes to check your answers with a friend.
II. Solve for the Unknown Part of a Given Proportion Using Mental Math
In the last section on unit rates and rates, you practiced figuring out a rate or a unit rate. Essentially, there was a missing part of the rate that you had to figure out. Here is an example of a problem like that.
Example
\begin{align*}\frac{25\ campers}{1\ tent} = \frac{75\ campers}{?\ tents}\end{align*}
Here we have a unit rate of twenty-five campers for one tent. Then we have another rate that says that we have 75 campers and we are trying to figure out how many tents are needed for the 75 campers.
These two ratios are equal and they form a proportion.
How can we figure out the missing number of tents?
We can use equal ratios to do this, or we can simply examine the problem and use mental math.
Sometimes, it makes more sense to simply figure out an answer in your head!
Our answer is three tents.
Often we can use mental math to quickly figure out the missing part of a proportion. We call this “solving a proportion.”
Example
\begin{align*}\frac{4}{8} = \frac{x}{16}\end{align*} We can think, “Four is half of eight, what is half of sixteen?”
Our answer is eight.
Practice using mental math to solve the following proportions.
- \begin{align*}\frac{1}{4} = \frac{x}{16}\end{align*}
- \begin{align*}\frac{3}{9} = \frac{x}{18}\end{align*}
- \begin{align*}\frac{5}{15} = \frac{1}{x}\end{align*}
Take a few minutes to check your answers with a partner. Did you both have the same answers for \begin{align*}x\end{align*} ?
III. Solve for the Unknown Part of Given Proportions Using Related Equations
Sometimes it can be very challenging to figure out the missing part of a proportion by using mental math. When this happens, you will need another strategy to figure out the missing part of the proportions. We can use cross products and simple equations to solve proportions.
How do we solve a proportion using cross products?
To understand how this works, let’s look at an example.
Example
Tony swims 10 laps in 30 minutes. How long does it take him to swim 15 laps?
Our first step is to write two ratios.
\begin{align*}\frac{10}{30}\end{align*} This is our known information.
\begin{align*}\frac{15}{x}\end{align*} This is what we are trying to figure out.
Notice that we put the same unit in the numerator of both ratios and the same unit in the denominator of both units.
\begin{align*}\frac{laps}{\text{min}\ utes} = \frac{laps}{\text{min}\ utes}\end{align*}
Now we can write a proportion.
\begin{align*}\frac{10}{30} = \frac{15}{x}\end{align*}
Our answer is not obvious in this problem. Because of this, we need to use cross products. We multiply 10 times \begin{align*}x\end{align*} and get \begin{align*}10x\end{align*} and then we multiply 15 times 30 and get 450.
\begin{align*}10x = 450\end{align*}
We can ask ourselves, “what times ten will give me 450?” or we can simplify the zeros and solve.
\begin{align*}1x = 45\end{align*} Here is our answer if we simplify the zeros. 1 times 45 equals 45.
Or we can think “10 times 45 equals 450.”
Our answer is 45.
Writing an equation with an unknown quantity can give you a different perspective on the problem. Often we can simplify or use mental math once we see the problem written in a new way.
Real Life Example Completed
Stocking Shelves
Use your knowledge of proportions to solve the following problem. Reread the problem and underline the important information to start.
Chase, Marc and Kris are all working at the supermarket as stock guys. After school, they work to stock the shelves at the local supermarket with all kinds of cans. In fact, they have gotten quite fast at it and they love to have contests to see who is the fastest. Often the one who loses has to treat the ice cream after work.
It takes Chase 15 minutes to stock three shelves of canned goods, and it takes Marc 45 minutes to stock nine shelves.
“I am definitely faster,” Chase tells Marc one afternoon.
“I don’t think so, I think we are both the same.” Marc disagrees.
Chase and Marc continue to argue. Who is correct?
Their friend Kris stocks shelves at the same rate as Chase does. If he stocks 12 shelves at this rate, how many minutes does it take him?
The first problem is to figure out which boy is faster at stocking shelves. To do this, we need to write two ratios and see if they form a proportion. If they form a proportion, then Marc is correct.
\begin{align*}\frac{15}{3} = \frac{45}{9}\end{align*}
If we simplify these ratios, we end up with \begin{align*}\frac{5}{1}\end{align*}. The ratios form a proportion, so Marc is correct. The boys both work at the same pace.
Now Kris works at the same rate as Chase (Marc too). He stocked 12 shelves. Given the rate, how long did it take him?
Once again, we need to write two ratios to form a proportion. The unknown will be the time that it takes Kris.
\begin{align*}\frac{15}{3} = \frac{x}{12}\end{align*}
If we look at this proportion and ask ourselves, “What times 3 equals twelve?” The answer is four. We can use that to form an equal ratio.
15 \begin{align*}\times\end{align*} 4 \begin{align*}=\end{align*} 60.
It takes Kris 60 minutes to stock his twelve shelves. All three boys are equal in their rate of speed. They decide to take turns buying the ice cream.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Proportion
- two equal ratios.
- Ratio
- a comparison of two quantities can be written in fraction form, with a colon or with the word “to”.
- Cross Products
- to multiply the diagonals of each ratio of a proportion.
Technology Integration
Khan Academy, Solving Proportions
James Sousa, Introduction to Proportions
James Sousa, Solving Basic Proportions
James Sousa, Applications Using Proportions
Other Videos:
http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.RATE&lesson=html/video_interactives/rateRatioProportions/rateRatioProportionsSmall.html – A video applying photography to rate, ratio and proportion.
Time to Practice
Directions: Use cross products or simplifying to identify whether each pair of ratios form a proportion. If they do, write yes. If not, write no.
1. \begin{align*}\frac{1}{2} = \frac{6}{12}\end{align*}
2. \begin{align*}\frac{1}{3} = \frac{4}{12}\end{align*}
3. \begin{align*}\frac{1}{4} = \frac{3}{15}\end{align*}
4. \begin{align*}\frac{5}{6} = \frac{10}{12}\end{align*}
5. \begin{align*}\frac{3}{4} = \frac{6}{10}\end{align*}
6. \begin{align*}\frac{2}{5} = \frac{6}{15}\end{align*}
7. \begin{align*}\frac{2}{7} = \frac{4}{21}\end{align*}
8. \begin{align*}\frac{4}{7} = \frac{12}{21}\end{align*}
9. \begin{align*}\frac{7}{8} = \frac{14}{16}\end{align*}
10. \begin{align*}\frac{25}{75} = \frac{1}{3}\end{align*}
11. \begin{align*}\frac{11}{33} = \frac{1}{3}\end{align*}
12. \begin{align*}\frac{15}{33} = \frac{2}{3}\end{align*}
Directions: Use mental math to solve the unknown part of each proportion.
13. \begin{align*}\frac{1}{2} = \frac{x}{8}\end{align*}
14. \begin{align*}\frac{1}{2} = \frac{5}{x}\end{align*}
15. \begin{align*}\frac{1}{3} = \frac{4}{x}\end{align*}
16. \begin{align*}\frac{2}{3} = \frac{x}{6}\end{align*}
17. \begin{align*}\frac{1}{2} = \frac{x}{16}\end{align*}
18. \begin{align*}\frac{5}{6} = \frac{x}{12}\end{align*}
19. \begin{align*}\frac{14}{16} = \frac{x}{8}\end{align*}
20. \begin{align*}\frac{1}{2} = \frac{x}{18}\end{align*}
21. \begin{align*}\frac{1}{4} = \frac{x}{20}\end{align*}
22. \begin{align*}\frac{1}{4} = \frac{x}{24}\end{align*}
23. \begin{align*}\frac{1}{4} = \frac{x}{40}\end{align*}
24. \begin{align*}\frac{2}{4} = \frac{x}{40}\end{align*}
25. \begin{align*}\frac{25}{50} = \frac{2}{x}\end{align*}