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# 4.1: Ratios and Rates

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Top of the World - Mount Everest

Josh sat at the breakfast table with his nose in a book. His Mom poured some cereal into a bowl and put it down in front of him. Karen, Josh’s sister came into the kitchen and sat down next to him.

“I am reading a book about Mount Everest,” Josh said without looking up.

“Alright Josh, now put the book down and eat something,” their Mom instructed.

“It is terrific. Do you know that during the best year on Everest, and best meaning the year that the least number of people died, that 129 people summitted and only 8 died,” Josh said smiling.

“That hardly seems like something to smile about,” Mom said sipping her coffee.

“Yeah, how morbid,” Karen chimed in.

“Listen it may be morbid, but it is a fact. During the worst year, only 98 people summitted and 15 died. That is a lot of people. I mean we can compare the number of people who summitted and the number of people who didn’t make it,” Josh said.

“Well, it proves that Everest is a dangerous place and not a trip to be taken lightly,” Mom said.

“Yes, but think about how amazing it would be to stand on the top of the world!” Josh said.

“How do they figure out that the best year is the best and the worst year is the worst based on those numbers?” Karen asked.

“They compare the ratios,” Josh explained. “Let me tell you how.”

Do you know how? Given this information, Josh is going to show Karen how to simplify and compare ratios. In this lesson, you will learn how to write ratios and how to understand them better. Pay close attention and you can work with the ratios of the best and worst years on Everest at the end of the lesson.

What You Will Learn

In this lesson, you will learn how to execute the following skills.

• Write, compare and order ratios.
• Find and apply unit rates.
• Find and apply equivalent rates.
• Solve real-world problems involving ratios and rates.

Teaching Time

I. Write, Compare and Order Ratios

A ratio is a comparison of two things or two quantities. The key thing to notice about ratios is that we are comparing. There are many different ways to compare things in mathematics. For example, imagine that there are 25 students in a class; 12 boys and 13 girls. You could say that the ratio of boys to girls is 12 to 13. You could also write this ratio as 12 : 13 or 1213\begin{align*}\frac{12}{13}\end{align*}. The ratio of boys to total students can be written as 12 to 25, 12 : 25, or 1225\begin{align*}\frac{12}{25}\end{align*}. The ratio of girls to total students can be written as 13 to 25, 13 : 25, or 1325\begin{align*}\frac{13}{25}\end{align*}. You will also see that the order that you compare makes a difference. If you are comparing boys to total students, then the number of boys comes first or if you are using fraction form, it is the numerator.

That is a great question. There are three different ways to write ratios. You can write them with a colon between the two values that you are comparing, you can write them using the word “to”, and you can write them by putting the values in fraction form. You can choose which way you want to write a ratio and these ways are interchangeable too.

Take a minute to write the definition of a ratio and the three ways to write a ratio in your notebook.

We just looked at the three ways to write a ratio when you are making a simple comparison. Remember to read the information carefully so that you are clear what is being compared. Now let’s look at an example where you might need to figure something out to write the ratio.

Example

There are 32 red and yellow candies in a bag. There are 10 yellow candies. What is the ratio of red candies to total candies in the bag?

We need the ratio of red candies to total candies. We know that there are 32 total candies. We need to find the number of red candies in the bag before writing the ratio.

3210=22\begin{align*}32-10 = 22\end{align*}

Now write the ratio of red candies to total candies. Because a ratio is a comparison, it can be simplified. Make sure to reduce the ratio to lowest terms. This makes it easier to understand the quantities that are being compared.

red candiestotal candies=2232=1116\begin{align*}\frac{\text{red candies}}{\text{total candies}} = \frac{22}{32} = \frac{11}{16}\end{align*}

We also could have written this ratio in two other ways.

22 to 32 then simplified to 11 to 16

OR

22 : 32 then simplified to 11 : 16

All of these answers would have been correct. Remember that you can interchange the form that you choose to write a ratio.

We can also compare ratios. This is when we have two or more ratios and we want to figure out which ones are larger and which ones are smaller. Let’s look at an example.

Example

Mr. Collison’s class has 30 total students. Of these, 12 are boys. Mrs. Peterson’s class has 25 students. Of these, 11 are boys. Which class has a higher ratio of boys to total students?

First, find the ratio of boys to total students for both classes. You need to do this first because these are the quantities that you are comparing.

Mr. Collison’s class: 1230=25\begin{align*}\frac{12}{30} = \frac{2}{5}\end{align*}

Mrs. Peterson’s class: 1125\begin{align*}\frac{11}{25}\end{align*}

Now compare the ratios the same way you compare fractions. Find a common denominator and compare the numerators.

The least common denominator is 25.

25(55)=1025\begin{align*}\frac{2}{5} \left(\frac{5}{5} \right) = \frac{10}{25}\end{align*}

1025<1125\begin{align*}\frac{10}{25} < \frac{11}{25}\end{align*}, since 10 is less than 11.

Mrs. Peterson’s class has a higher ratio of boys to total students.

Note: In Example 2, even though Mr. Collison’s class had a larger number of boys in the class, Mrs. Peterson’s class had a larger ratio of boys to total students.

We can also order ratios. When you have more than two ratios, you can write them in order from least to greatest or from greatest to least. Let’s look at an example.

Example

Order the following ratios from least to greatest: 10 to 15, 1636\begin{align*}\frac{16}{36}\end{align*}, 12 : 48

The first thing to notice is that these ratios are all in different forms. Let’s write them in the same form first of all. Let’s work with fraction form so that we can apply what we know about comparing and ordering fractions.

10 to 15=1015\begin{align*}10 \ \text{to} \ 15 = \frac{10}{15}\end{align*}

1636\begin{align*}\frac{16}{36}\end{align*}

12:48=1248\begin{align*}12:48 = \frac{12}{48}\end{align*}

Now notice that none of these fractions are in simplest form. We can simplify them and that will make it much easier to order them.

101516361248=23=49=14\begin{align*}\frac{10}{15} &= \frac{2}{3}\\ \frac{16}{36} &= \frac{4}{9}\\ \frac{12}{48} &= \frac{1}{4}\end{align*}

If you understand fractions, we can simply order them right now. We know that one-fourth is the smallest part. That four is almost half of nine, so that would be the middle value, and that two-third would be the greatest part.

The answer is 14,49,23\begin{align*}\frac{1}{4}, \frac{4}{9}, \frac{2}{3}\end{align*}.

Well in that case, you could rewrite them all using a common denominator. Then you will be able to order the numerators. Let’s take a look at that.

144923=936=1636=2436\begin{align*}\frac{1}{4} &= \frac{9}{36}\\ \frac{4}{9} &= \frac{16}{36}\\ \frac{2}{3} &= \frac{24}{36}\end{align*}

You can see that our original work was correct.

II. Find and Apply Unit Rates

A unit rate is a special kind of ratio, where the second number, or the denominator, is equal to one. With a unit rate, you are comparing a quantity to one. Some common examples of unit rates are miles per gallon, price per pound, and pay rate per hour.

To find a unit rate, simplify the ratio so that you have a 1 in the denominator. You can simply divide the first number in the ratio by the second. Make sure you keep track of the units.

Write this information on unit rates down in your notebook. Then continue with the example.

Example

Kayla bought 5.5 pounds of apples. She paid a total of $7.15. What was the unit rate of the apples per pound? A key word is here “per pound”. When you see the word “per”, you should know that you are working with unit rates. You need to find the price per pound. We can write the ratio of price to pounds using the information in the problem. That is what we are comparing, so that is how we write the ratio. Then we can fill in the given information. pricepounds=$7.155.5 pounds\begin{align*}\frac{price}{pounds} = \frac{\ 7.15}{5.5 \ pounds}\end{align*}

Now divide to find the price per pound. We divide the price by the number of pounds that she bought.

5.5)7.15¯¯¯¯¯¯¯¯¯¯¯¯\begin{align*}{5.5 \overline{){7.15\; }}}\end{align*}

$7.155.5 pounds=$1.301 pound\begin{align*}\frac{\7.15}{5.5 \ pounds} = \frac{\ 1.30}{1 \ pound}\end{align*}

The unit rate is $1.30 per pound. Example Brian worked for 8 hours yesterday and made a total of$86. What is his pay rate?

Write a ratio comparing pay rate to hours worked. The rate that we are looking for is what Brian made “per” hour. Even though the problem doesn’t use the word “per” a pay rate is per hour.

pay ratehours worked=868 hours\begin{align*}\frac{\text{pay rate}}{\text{hours worked}} = \frac{\ 86}{8 \ hours}\end{align*} Now divide to find the unit rate.868 hours=10.751 hour\begin{align*}\frac{\86}{8 \ hours} = \frac{\ 10.75}{1 \ hour}\end{align*} Brian’s pay rate is10.75 per hour.

You may also see rates that are equal. Let’s look at equivalent rates.

III. Find and Apply Equivalent Rates

An equivalent rate is a rate that is equal but is written in a different form.'

To find an equivalent rate, you need to write a fraction that is equivalent to the given fraction, or rate.

Example

Find an equivalent rate for this comparison.

2.001=?8\begin{align*}\frac{\2.00}{1} = \frac{?}{8}\end{align*} Notice that we have a unit rate here. We know that the unit rate is two dollars for every one thing. We want eight. We can figure out how to work on this problem by thinking mathematically. 1×8=8\begin{align*}1 \times 8 = 8\end{align*} Just like when we worked with fractions, whichever operation we perform with the denominator must be performed with the numerator too. We multiplied by eight, so we need to do that with the numerator too.2.001=16.008\begin{align*}\frac{\2.00}{1} = \frac{\ 16.00}{8}\end{align*} These two rates are equivalent. As long as you multiply the numerator and the denominator by the same value, you will always create an equivalent rate! IV. Solve Real – World Problems Involving Ratios and Rates Now that you understand ratios and rates, we can look at applying this information when working with real – world problems. Let’s look at some examples. Example Myra’s team scored 10 goals in the last 3 games. At this rate, how many goals will Myra’s team score in 6 games? First, write the ratio to show the team’s scoring rate. goalsgames=10 goals3 games\begin{align*}\frac{goals}{games} = \frac{10 \ goals}{3 \ games}\end{align*} You need to know the equivalent rate for 6 games. Notice that the second ratio has the games in the same spot of the denominator. Be sure that you write the ratios so that the same quantities are being compared. If you mix them up, you get a different result. 10 goals3 games=? goals6 games\begin{align*}\frac{10 \ goals}{3 \ games} = \frac{? \ goals}{6 \ games}\end{align*} Look at the two fractions. The denominator is doubled in the second fraction. So, multiply the first fraction by an equivalent of 1 in order to get the second fraction. 10 goals3 games(22)=20 goals6 games\begin{align*}\frac{10 \ goals}{3 \ games} \left(\frac{2}{2}\right) = \frac{20 \ goals}{6 \ games}\end{align*} An equivalent rate is 20 goals in 6 games. So at this rate, Myra’s team will score 20 goals. Example A store sells salmon for6.99 per pound. What is the rate for 6 pounds of salmon?

You know the rate per pound. You need to find the rate for 6 pounds. So you can multiply the unit rate by 6 to find the equivalent ratio.

$6.991 pound(66)=$41.946 pounds\begin{align*}\frac{\6.99}{1 \ pound} \left(\frac{6}{6} \right) = \frac{\41.94}{6 \ pounds}\end{align*}

The rate for 6 pounds of salmon is 41.94. Now let’s go back to the problem from the introduction and work on solving that problem. ## Real-Life Example Completed The Top of the World - Mount Everest Here is the original problem once again. First, reread it. Then you will need to write and simplify two ratios. One ratio will represent the numbers from the best year on Everest and one ratio will represent the numbers from the worst year on Everest. There are two parts to your answer. Josh sat at the breakfast table with his nose in a book. His Mom poured some cereal into a bowl and put it down in front of him. Karen, Josh’s sister came into the kitchen and sat down next to him. “What are you reading?” she asked. “I am reading a book about Mount Everest,” Josh said without looking up. “Alright Josh, now put the book down and eat something,” their Mom instructed. “It is terrific. Do you know that during the best year on Everest, and best meaning the year that the least number of people died, that 129 people summitted and only 8 died,” Josh said smiling. “That hardly seems like something to smile about,” Mom said sipping her coffee. “Yeah, how morbid,” Karen chimed in. “Listen it may be morbid, but it is a fact. During the worst year, only 98 people summitted and 15 died. That is a lot of people. I mean we can compare the number of people who summitted and the number of people who didn’t make it,” Josh said. “Well, it proves that Everest is a dangerous place and not a trip to be taken lightly,” Mom said. “Yes, but think about how amazing it would be to stand on the top of the world!” Josh said. “How do they figure out that the best year is the best and the worst year is the worst based on those numbers?” Karen asked. “They compare the ratios,” Josh explained. “Let me tell you how.” Now write your ratios. Remember to simplify them. There are two parts to your answer. Solution to Real – Life Example We are going to write two ratios that we can compare. One ratio will represent the best year on Everest and one will represent the worst year on Everest. First, let’s look at the numbers from the best year on Everest. 129 people summitted 8 died The ratio is 129 : 8. However, we want to simplify this ratio to get a better idea of the size of the ratio. Let’s rewrite it in fraction form and simplify it. 1298=161\begin{align*}\frac{129}{8}= \frac{16}{1}\end{align*} This means that for every 16 people who summitted one did not make it. Now let’s look at the numbers for the worst year on Everest. 98 people summitted 16 people died 9816=6.5 to 1\begin{align*}\frac{98}{16} = 6.5 \ to \ 1\end{align*} This means that for every 612\begin{align*}6 \frac{1}{2}\end{align*} people who summitted, one did not make it. From these ratios, you can see how the best year compares to the worst year on Everest. ## Vocabulary Here are the vocabulary words that are found in this lesson. Ratio a way of comparing two numbers or quantities. Ratios can be written in fraction form, with a colon or by using the word “to”. Unit Rate a ratio that is comparing a quantity to one. The word “per” is a key word with unit rates. Equivalent Rate two rates that are equal although different values are being used to represent the same quantities. Resources www.mnteverest.net ## Time to Practice Directions: Look at each ratio. Then write it in the other two ways that it can be written. 1. 16 to 3 2. 4 to 5 3. 1 : 4 4. 121\begin{align*}\frac{12}{1}\end{align*} 5. 6 : 11 6. 33 to 100 7. 49\begin{align*}\frac{4}{9}\end{align*} 8. 3 to 4 9. 45 to 12 10. 12 : 12 Directions: Simplify each ratio and write your answer in fraction form. 1. 4 to 12 2. 5 : 20 3. 36 to 6 4. 18 : 36 5. 20 to 100 Directions: Use what you have learned about ratios to solve each problem. In Kyle’s drawer, there are 14 pairs of white socks and 8 pairs of black socks 1. Write the ratio of black socks to white socks. 2. Write the ratio of black socks to total socks. 3. Write the ratio of white socks to total socks. 4. There are 150 apartments in the Gray building. Of these, 60 are rented and the rest are owned. There are 65 apartments in the Black building. Of these, 45 are rented and the rest are owned. Which building has a larger ratio of owned apartments to total apartments, and by how much? 5. Susan drove a total of 182 miles in three and a quarter hours. If she drove the same speed the whole drive, what was her rate in miles per hour? 6. Holly works at a library reshelving books. She reshelved 960 books in 4 hours. What is Holly’s rate of reshelving in books per hour? 7. Sam bought 9.5 pounds of peaches to make a pie. The peaches cost15.39. What was the unit rate of the peaches?
8. A vacation house costs \$1,100 to rent per month. If there are 30 days in the month, what is the daily rate?
9. Julie can type 55 words per minute. What is the equivalent rate of typing 15 minutes?
10. Don can wrap 8 presents in an hour. What is Don’s rate for 12 presents?

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