4.1: Write, Compare, and Order Ratios
Have you ever wondered about Mount Everest? Take a look at this dilemma.
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.”
Do you know how? Given this information, Josh is going to show Karen how to simplify and compare ratios. You can learn all about ratios in this Concept.
Guidance
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.
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 \begin{align*}\frac{12}{13}\end{align*}
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 a situation where you might need to figure something out to write the ratio.
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.
\begin{align*}3210 = 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.
\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. Take a look.
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: \begin{align*}\frac{12}{30} = \frac{2}{5}\end{align*}
Mrs. Peterson’s class: \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.
\begin{align*}\frac{2}{5} \left(\frac{5}{5} \right) = \frac{10}{25}\end{align*}
\begin{align*}\frac{10}{25} < \frac{11}{25}\end{align*}
Mrs. Peterson’s class has a higher 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.
Order the following ratios from least to greatest: 10 to 15, \begin{align*}\frac{16}{36}\end{align*}
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.
\begin{align*}10 \ \text{to} \ 15 = \frac{10}{15}\end{align*}
\begin{align*}\frac{16}{36}\end{align*}
\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.
\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 onefourth is the smallest part. That four is almost half of nine, so that would be the middle value, and that twothird would be the greatest part.
The answer is \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.
\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.
Now simplify each ratio.
Example A
\begin{align*}\frac{7}{21}\end{align*}
Solution: \begin{align*}\frac{1}{3}\end{align*}
Example B
\begin{align*}5:30\end{align*}
Solution: \begin{align*}1:6\end{align*}
Example C
\begin{align*}24\end{align*}
Solution: \begin{align*}2\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
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.
\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
\begin{align*}\frac{98}{16} = 6.5 \ to \ 1\end{align*}
This means that for every \begin{align*}6 \frac{1}{2}\end{align*}
From these ratios, you can see how the best year compares to the worst year on Everest.
Vocabulary
 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”.
Guided Practice
Here is one for you to try on your own.
Write a ratio in simplest form to describe this situation.
Marcy loves cookies. She ate 14 cookies in 28 minutes. Write a ratio to compare cookies to minutes.
Solution
First, we are going to write a ratio to compare cookies to minutes, so we will write the number of cookies first.
\begin{align*}14\end{align*}
The number of minutes comes second.
\begin{align*}14:28\end{align*}
We could also write this ratio in fraction form.
\begin{align*}\frac{14}{28}\end{align*}
Now let's simplify it.
\begin{align*}\frac{14}{28} = \frac{1}{2}\end{align*}
This is our answer.
Video Review
Khan Academy Ratios in Simplest Form
Practice
Directions: Look at each ratio. Then write it in the other two ways that it can be written.
 16 to 3
 4 to 5
 1 : 4

\begin{align*}\frac{12}{1}\end{align*}
121  6 : 11
 33 to 100

\begin{align*}\frac{4}{9}\end{align*}
49  3 to 4
 45 to 12
 12 : 12
Directions: Simplify each ratio and write your answer in fraction form.
 4 to 12
 5 : 20
 36 to 6
 18 : 36
 20 to 100
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Here you'll learn to write, compare and order ratios.