Lesson 10: Comparing Situations by Examining Ratios

Let’s use ratios to compare situations.

Mai and Jada each ran on a treadmill. The treadmill display shows the distance, in miles, each person ran and the amount of time it took them, in minutes and seconds.

Here is Mai’s treadmill display:

1. What is the same about their workouts? What is different about their workouts?
2. If each person ran at a constant speed the entire time, who was running faster? Explain your reasoning.

10.2: Concert Tickets

Diego paid \$47 for 3 tickets to a concert. Andre paid \$141 for 9 tickets to a concert. Did they pay at the same rate? Explain your reasoning.

GeoGebra Applet ycJSQpXT

10.3: Sparkling Orange Juice

Lin makes sparkling orange juice by mixing 3 liters of orange juice with 4 liters of soda water. Noah makes sparkling orange juice by mixing 4 liters of orange juice with 5 liters of soda water. How do the two mixtures compare in taste? Explain your reasoning. If you get stuck, you can draw double number line diagrams to represent each situation.

GeoGebra Applet ycJSQpXT

GeoGebra Applet ycJSQpXT

Summary

Sometimes we want to know whether two situations are described by the same rate. To do that, we can write an equivalent ratio for one or both situations so that one part of their ratios has the same value. Then we can compare the other part of the ratios.

For example, do these two paint mixtures make the same shade of orange?

• Kiran mixes 9 teaspoons of red paint with 15 teaspoons of yellow paint.
• Tyler mixes 7 teaspoons of red paint with 10 teaspoons of yellow paint.

Here is a double number line that represents Kiran's paint mixture. The ratio $9:15$ is equivalent to the ratios $3:5$ and $6:10$.

For 10 teaspoons of yellow paint, Kiran would mix in 6 teaspoons of red paint. This is less red paint than Tyler mixes with 10 teaspoons of yellow paint. The ratios $6:10$ and $7:10$ are not equivalent, so these two paint mixtures would not be the same shade of orange.

When we talk about two things happening at the same rate, we mean that the ratios of the quantities in the two situations are equivalent. There is also something specific about the situation that is the same.

• If two ladybugs are moving at the same rate, then they are traveling at the same constant speed.
• If two bags of apples are selling for the same rate, then they have the same unit price.
• If we mix two kinds of juice at the same rate, then the mixtures have the same taste.
• If we mix two colors of paint at the same rate, then the mixtures have the same shade.

Glossary

same rate

same rate

In two situations involving ratios of the same two quantities, if the ratio of the quantities in one situation is equivalent to the ratio of the quantities in the other situation then we say the two situations involve the same rate.