## Table of Contents

- What Are Equivalent Ratios?
- How to Find Equivalent Ratios
- Equivalent Ratios Practice
- Equivalent Ratios Uses
- Lesson Summary

Learn about equivalent ratios, including the equivalent ratios definition and examples. Learn how to find equivalent ratios and where to put these skills into use.
Updated: 05/09/2021

- What Are Equivalent Ratios?
- How to Find Equivalent Ratios
- Equivalent Ratios Practice
- Equivalent Ratios Uses
- Lesson Summary

A **ratio** is a mathematical way of comparing two quantities. The first quantity to be compared is written first, and the second quantity to be compared is written second. For example, at 8 AM, there are 5 red cars and 2 blue cars in a parking lot. In this case, the ratio of red cars to blue cars is 5 to 2. To express this in ratio form, it can be written in two different ways, 5:2 or 5/2. The ratio of blue cars to red cars is 2 to 5, which can be written 2:5 or 2/5.

At 5 PM, there are 10 red cars and 4 blue cars, so the ratio of red cars to blue cars is 10:4 or 10/4. Comparing this to the ratio in the morning, we can say that in mathematical terms the ratio 5/2 and 10/4 are equivalent ratios. This is because the values of the ratios are equal. You can derive 10/4 from 5/2 by dividing both numbers by 2. And you can derive 5/2 from 10/4 by multiplying both numbers by 2. In this case the ratios changed in proportion, as reflected in the following equivalent ratios table..

**Equivalent ratios** are fractions that are derived from each other so that their values are equal. They can be considered as ratios that compare with each other such that one ratio is equal in value to the other ratio. The ratios are considered to be equivalent because the numerator and denominator are changed in proportion. Equivalent ratios are multiples or factors of each other.

Two ratios are equivalent if:

- When multiplying the numerator and denominator of one ratio by the same number you get the other ratio
- When dividing the numerator and denominator of one ratio by the same number you get the other ratio

Now, suppose in the evening there were 10 red cars and 3 blue cars. The ratios 5/2 and 10/3 are NOT equivalent because you cannot derive the second ratio by multiplying both numbers of the first ratio by the name number. In this case the ratios did NOT change in proportion.

Let's look at some equivalent ratios examples. Consider a parking lot with 1 bicycle and 1 car and another parking lot with 3 bicycles and 3 cars. As the number of bicycles and cars triples, the number of wheels for the bicycles and the cars also triples. One bicycle has 2 wheels, 3 bicycles have 6 wheels. One car has 4 wheels, 3 cars have 12 wheels. The ratio of wheels for 1 bicycle to 1 car is 2/4. And the ratio of wheels for 3 bicycles to 3 cars is 6/12. We can say that 6/12 and 2/4 are **equivalent ratios** because when the number of bicycles tripled, the number of cars also tripled.

In mathematical terms, we can find the following pattern: when the numerator and denominator of the first ratio are multiplied by the same number 3, you get the second ratio. And when the numerator and denominator of the second ratio are divided by the same number 3, you get the first ratio.

The two ratios 2/4 and 6/12 are **equivalent ratios** because they represent the same value.

To find an equivalent ratio, multiply the numerator and denominator of the ratio by the same number OR divide the numerator and denominator of the ratio by the same number.

Given a ratio 3:5, how can you find an equivalent ratio? The first step would be to write the ratio in the form of a fraction 3/5. Now you can find any number of equivalent ratios by multiplying the numerator and denominator by the same number.

- 3/5 = 6/10 (multiply the numerator and denominator by 2)
- 3/5 = 9/15 (multiply the numerator and denominator by 3)
- 3/5 = 30/50 (multiply the numerator and denominator by 10)
- 3/5 = 1.5/2.5 (divide the numerator and denominator by 2)
- 3/5 = 0.3/0.5 (divide the numerator and denominator by 10)

You can also find equivalent ratios from other equivalent ratios. In this example, we already have 3/5 = 30/50, so you can multiply the numerator and denominator of the equivalent ratio 30/50 by the same number 3 to get another equivalent ratio:

- 30*3 / 50*3 = 90/150

The following ratios are all equivalent ratios to each other.

- 3/5 = 6/10 = 9/15 = 30/50 = 0.3/0.5 = 90/150

You can continue this way any number of times to get an infinite number of equivalent ratios.

To summarize, the method used to find equivalent ratios or equivalent fractions is as follows:

- Take the first ratio.
- Multiply OR divide the numerator and denominator by the same number to get the second ratio.
- Now take the first ratio OR the new equivalent ratio.
- Multiply OR divide the numerator and denominator by the same number to get a new equivalent ratio.
- Continue the process to find any number of equivalent ratios.

What if the ratio consists of larger numbers? Given the ratio 258:412, how will you find two equivalent ratios? The simplest way is to use a calculator. Write this ratio as a fraction, then divide the numerator and denominator by the same number. You can use any number as long as the same number is used to multiply or divide both the numerator and the denominator.

Start with multiplying the numerator and denominator by a small number, say 5. Input into the calculator the following and write down the answers.

- 258 x 5 = 1,290
- 412 x 5 = 2,060

The new ratio that is equivalent to 258/412 is 1,290/2,060.

What if you divide the first ratio by 2?

You will now get 129/206

Now we have 3 equivalent ratios:

258/412 = 1,290/2,060 = 129/206

If you want to find a ratio that is 33 times the multiple of the second ratio, you would multiply the numerator and denominator of your second ratio by 33 using your calculator. Then you will have:

258/412 = 1,290/2,060 = 129/206 = 42,570/67,980

Equivalent ratios have many different uses in everyday life. Here are some examples of how they are used.

Business: In a period of 20 minutes, a fast-food restaurant serves 5 people inside and 15 people in the drive-through. This can be written as a ratio 5:15. If they serve at the same rate, after 40 minutes the ratio of people served inside to the people in the drive-through will be 10:30. This is an equivalent ratio because as the number of minutes doubles, the number of people served inside and the number of people served outside also doubles. The two quantities of the first ratio 5:15 are multiplied by the same number 2 to get the second ratio 10:30.

Food: A recipe uses 2 cups of sugar and 1 stick of butter to make 20 cookies. What if you wanted to make only 10 cookies? Since the number of cookies is halved, you would half the amount of sugar and butter also. The ratio of sugar to butter for 20 cookies is 2/1. So, for 10 cookies you would use 1 cup of sugar and 0.5 stick of butter. This is an equivalent ratio because you divide the numerator and denominator of 2/1 by the same number 2 to get the new ratio 1/0.5.

Education: You complete one lab experiment in one week. This can be written as a ratio 1 to 1. What is the ratio of labs to weeks in 52 weeks? This is the ratio 52 to 52. These ratios are equivalent because you can divide both numbers of the second ratio by 52 to get the first ratio.

A **ratio** is a mathematical way of comparing two quantities. **Equivalent ratios** are ratios that are equal in value to each other. An equivalent ratio can be created by

- multiplying both numbers by the same number, OR
- dividing both numbers by the same number

After creating the first equivalent ratio, any number of new equivalent ratios can be created by applying one of the following methods:

- derive equivalent ratios from the first ratio
- derive equivalent ratios from one of the new equivalent ratios

Equivalent ratios have multiple uses in everyday life. We can find examples in business, education, technology, food, medicine and in many other areas. Equivalent ratios can be used in any situation in which quantities change in proportion to each other.

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Additional Activities

Equivalent ratios are fractions used to convert between units by representing equal measures. The exercises below will be used to assist you in further understanding the use of equivalent ratios and their role in conversions. Solutions are provided below

1. Are the fractions 3/20 and 9/60 equivalent?

2. What is the numerator of 3/65? What is the denominator?

3. If John takes 4 hours to do his homework, how many minutes did he take? What is the equivalent ratio to convert between minutes and hours?

4. Setup and solve the equivalent ratio for the following scenario. If Jamie was going on a road trip across the country and drove at a constant pace of 75 miles per hour, how many hours did it take him to travel 975 miles?

5. Setup and solve the equivalent ratio for the following scenario. If Betty makes $10.50 an hour, how much money will she make after working 80 hours?

6. Is 2/9 equivalent to 6/20?

7. Setup and solve the equivalent ratio for the following scenario. Justin can eat ten mini cheeseburgers as one meal. If they cost $1.25, how much does his meal cost?

1. Yes, the 3/20 is a reduced form of 9/60

2. The numerator is 3, the denominator is 65

3. It will take him 240 minutes, 1 hour = 60 minutes so the ratio is 1/60

4. 75 mi = 1 hr, 975 mi = 13 hours

5. 1 hr = $10.50, 80 hrs = $840

6. No

7. $1.25 = 1 cheeseburger, 10 cheeseburgers = $12.50

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