5.1: Ratios
Introduction
Rapid Reading
Kayla and Torrey are both in Mrs. Henderson’s seventh grade Language Arts class. On the first day of school, Mrs. Henderson posed a reading challenge to the class. The challenge was to see how many books they can each read throughout the entire year. The books would be read in silent reading or outside of school and the honor system would be employed. Each student was to keep track of the title and author of each book as they read it. At the end of the year, they will celebrate their accomplishments with a pizza party and students can earn extra credit towards their final grade. While some students were hesitant to take on the challenge, all loved the promise of a pizza party and extra credit. Mrs. Henderson won them over.
Kayla and Torrey have been participating in the challenge since day one. By October, eight weeks into school, Kayla had already finished 6 books.
“I have six read,” Kayla said biting into her sandwich at lunch.
“I think I am reading at the same pace,” Torrey said. “After four weeks, I had already finished 3 books.”
“Are you sure,” Kayla asked.
“Well, I need to count, but I am pretty sure that we have both finished reading the same number of books.”
Is Torrey correct? Have the girls each read the same number of books now that 8 weeks have passed?
To figure this out, you will need to use equivalent ratios. Equivalent ratios are the topic of this first lesson, pay attention and you be able to figure out if Torey and Kayla have both read the same number of books at the end of the lesson.
What You Will Learn
In this lesson, you will learn the following skills:
- Identify and write different forms of equivalent ratios among rational numbers.
- Write ratios in simplest form.
- Write and compare ratios in decimal form.
- Solve real-world problems involving ratios among rational numbers.
Teaching Time
I. Identify and Write Different Forms of Equivalent Ratios among Rational Numbers
In math and in real-life, we compare things all the time. We look at what we have and what someone else has or we look at the difference between values and we compare them. Comparing comes very naturally. Using ratios comes naturally too, because ratios are a way that we can compare numbers and values.
What is a ratio?
A ratio compares two numbers or quantities called terms. For example, suppose there are 3 green (G) apples and 4 red (R) apples in this basket.
We can express the ratio of green apples to red apples in the basket as a fraction.
\begin{align*}\frac{green}{red} = \frac{3}{4}\end{align*}
We can also express this ratio in words, 3 to 4, or using a colon, 3:4.
The ratio above compares one part of the apples in the basket to another part. The ratio above compares the apples that are green to the apples that are red.
A ratio may also express a part to a whole. For example, we can express the ratio of green apples to the total number of apples in the basket.
\begin{align*}\frac{green}{total} = \frac{green}{green + red} = \frac{3}{3 + 4} = \frac{3}{7}\end{align*}
There are a total of 7 apples in the basket, so the ratio of green apples to total apples is 3 to 7 or 3:7.
Here are the three ways that we can write a ratio:
- In fraction form using a fraction bar
- Using the word “to”
- Using a colon:
Take a few minutes to write these three ways in your notebook. Then you can refer back to them when needed.
5A. Lesson Exercises
Look at the drawing and write each the ratio three different ways.
Compare stars to suns
Take a few minutes to check your answers with a friend.
Now that you know how to write a ratio, let's look at equivalent ratios.
A ratio shows the relationship between two quantities. Equivalent ratios can be used to show the same relationship between two quantities. Remember that the word “equivalent” means equal.
Because we can write ratios in fraction form, we can use what we know about finding equivalent fractions to help us identify equivalent ratios. Here is where simplifying fractions is going to help us. We can simplify ratios to discover equivalence just as we can simplify fractions.
Example
Determine if these two ratios are equivalent \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{10}{15}\end{align*}.
We can start by simplifying the larger of the two fractions. If it simplifies to the same number as the smaller fraction, then we know that the two ratios are equivalent.
\begin{align*}\frac{10 \div 5}{15 \div 5} &= \frac{2}{3}\\ \frac{2}{3} &= \frac{2}{3}\end{align*}
These are equal ratios.
Simplifying is one way to check for equivalence. We can also create equivalent ratios by multiplying just as we would make equivalent fractions.
Change \begin{align*}\frac{2}{3}\end{align*} to a ratio with 15 as the second term (the denominator).
Since \begin{align*}3 \times 5 = 15\end{align*}, multiply both terms of the ratio \begin{align*}\frac{2}{3}\end{align*} by 5.
\begin{align*}\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}\end{align*}
This shows that the ratio \begin{align*}\frac{2}{3}\end{align*} is equivalent to the ratio \begin{align*}\frac{10}{15}\end{align*}.
So, the two ratios listed above are equivalent.
Example
Determine if these two ratios are equivalent 7:6 and 13:12.
Rewrite the ratios as fractions \begin{align*}\frac{7}{6}\end{align*} and \begin{align*}\frac{13}{12}\end{align*}.
Change \begin{align*}\frac{7}{6}\end{align*} to a ratio with 12 as the second term.
Since \begin{align*}6 \times 2 = 12\end{align*}, multiply both terms of the ratio \begin{align*}\frac{7}{6}\end{align*} by 2.
\begin{align*}\frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12}\end{align*}
When the second term (the denominator) is 12, the equivalent ratio for \begin{align*}\frac{7}{6}\end{align*} is \begin{align*}\frac{14}{12}\end{align*}, not \begin{align*}\frac{13}{12}\end{align*}.
So, 7:6 and 13:12 are not equivalent ratios.
Another way to determine if two ratios are equivalent is to cross multiply the terms in the ratio. If the cross products are equal, then the two ratios are equivalent. If the cross products are not equal, then the two ratios are not equivalent.
Example
Determine if these two ratios are equivalent \begin{align*}\frac{4}{8}\end{align*} and \begin{align*}\frac{1}{2}\end{align*}.
We use the \begin{align*}\overset{?}{=}\end{align*} symbol to show that the two ratios below may or may not be equal.
\begin{align*}\frac{4}{8} \overset{?}{=} \frac{1}{2}\end{align*}
Now, we cross multiply. To cross multiply, multiply the circled pairs of numbers shown below. The product we get when we multiply each circled pair of numbers is called a cross product.
\begin{align*}8 \times 1 &\overset{?}{=} 4 \times 2\\ 8 &\overset{?}{=} 8\\ 8 &= 8\end{align*}
Since \begin{align*}8 = 8\end{align*}, the cross products are equal. This means that \begin{align*}\frac{4}{8} = \frac{1}{2}\end{align*}, and those two ratios are equivalent.
5B. Lesson Exercises
Determine whether or not each pair of ratios is equivalent. Write yes or no.
- \begin{align*}\frac{3}{4}\end{align*} and \begin{align*}\frac{9}{12}\end{align*}
- \begin{align*}\frac{5}{6}\end{align*} and \begin{align*}\frac{20}{30}\end{align*}
- \begin{align*}\frac{4}{5}\end{align*} and \begin{align*}\frac{8}{10}\end{align*}
Take a few minutes to check your work with a partner.
II. Write Ratios in Simplest Form
In the last section, we touched briefly on how to simplify ratios that are in fraction form to see if they are equivalent or not. We will explore this further in this section.
What does it mean to simplify?
To simplify means to make smaller. When we simplify, we make a fraction smaller. It is still equivalent to the larger form of the fraction, but it is simpler.
How do we simplify a ratio in fraction form?
To write a ratio in simplest form, find the greatest common factor of both terms in the ratio. The greatest common factor of two numbers is the greatest number that divides both numbers evenly. Then, divide both terms of the ratio by the greatest common factor.
This is basically the same procedure you use to rewrite a fraction in simplest form. Let’s look at an example.
Example
Write this ratio in simplest form \begin{align*}\frac{20}{24}\end{align*}.
First, find the greatest common factor of the terms 20 and 24.
The factors of 20 are 1, 2, 4, 5, 10, and 20.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The factors that both 20 and 24 have in common are 1, 2 and 4.
The greatest of those common factors is 4, so divide both terms by 4 to write the ratio in simplest form.
\begin{align*}\frac{20}{24} = \frac{20 \div 4}{24 \div 4} = \frac{5}{6}\end{align*}
So, the simplest form of the ratio \begin{align*}\frac{20}{24}\end{align*} is \begin{align*}\frac{5}{6}\end{align*}.
How is this useful when working with ratios?
It is useful because we can simplify ratios in fraction form to see if they are equivalent. We can also use a simplified ratio to find an equivalent ratio. To do this, you can multiply both terms of the original ratio by the same number to find an equivalent ratio.
Example
Write three equivalent ratios for this ratio 1:9.
The ratio 1:9 or \begin{align*}\frac{1}{9}\end{align*} is already in simplest form. Notice that we wrote the ratio into fraction form so that it is easier to work with.
Now, we will write three equivalent ratios by multiplying both terms by the same number.
It does not matter by which numbers we choose to multiply the terms. Let's multiply by 2, first.
\begin{align*}\frac{1}{9} = \frac{1 \times 2}{9 \times 2} = \frac{2}{18} \ \text{or} \ 2:18\end{align*}
Let's multiply by 5, next.
\begin{align*}\frac{1}{9} = \frac{1 \times 5}{9 \times 5} = \frac{5}{45} \ \text{or} \ 5:45\end{align*}
Let's multiply by 100, next.
\begin{align*}\frac{1}{9} = \frac{1 \times 100}{9 \times 100} = \frac{100}{900} \ \text{or} \ 100:900\end{align*}
So, three ratios that are equivalent to 1:9 are 2:18, 5:45, and 100:900.
5C. Lesson Exercises
- Simplify this ratio 4:16
- Simplify this ratio 3 to 18
- Write an equivalent ratio to 4:5
Take a few minutes to check your work with a friend. Are your answers accurate? Compare answers for number 3, you may each have a different answer for this one. Are they equivalent?
III. Write and Compare Ratios in Decimal Form
Just as fractions can be written in decimal form, ratios can be written in fraction form, so they can also be written as decimals. Now we will look at how to write a ratio as a decimal.
How do we write a ratio as a decimal?
To convert a ratio to decimal form, write the ratio as a fraction. Then divide the term above the fraction bar by the term below the fraction bar.
Let’s look at how to do this.
Example
Rewrite the ratio 1 to 4 in decimal form.
The ratio 1 to 4 can be expressed as the fraction \begin{align*}\frac{1}{4}\end{align*}. This is our first step.
Next, divide the term above the fraction bar, 1, by the term below the fraction bar, 4.
\begin{align*}\frac{1}{4} = {4 \overline{ ) 1 \;\;\;}}\end{align*}
Since 1 cannot be evenly divided by 4, rewrite 1 as a decimal with a zero after the decimal point You can do this because \begin{align*}1 = 1.0 = 1.00 = 1.000\end{align*}. Before you divide, write a decimal point in the quotient directly above the decimal point in the dividend. Then divide.
\begin{align*}& \overset{ \ \ 0.2}{4 \overline{ ) {1.0 \;}}}\\ & \quad \underline{-8}\\ & \quad \ \ 2 \end{align*}
Continue adding zeroes after the decimal point and diving until the quotient has no remainder.
\begin{align*}& \overset{ \ \ 0.25}{4 \overline{ ) {1.00 \;}}}\\ & \quad \underline{-8 \;\;}\\ & \quad \ \ 20\\ & \ \ \ \underline{-20}\\ & \qquad \ 0 \end{align*} The decimal form of the ratio \begin{align*}\frac{1}{4}\end{align*} is 0.25.
Example
Rewrite the ratio 9:5 in decimal form.
The ratio 9:5 can be expressed as the fraction \begin{align*}\frac{9}{5}\end{align*}.
Next, divide the term above the fraction bar, 9, by the term below the fraction bar, 5.
\begin{align*}& \overset{ \ \ \ 1}{5 \overline{ ) { \ 9 \;}}}\\ & \ \underline{-5\;}\\ & \ \ \ 4 \end{align*} There is a remainder. So, add zeroes after the decimal point in 9 to continue dividing.
\begin{align*}& \overset{ \ \ 1.8}{5 \overline{ ) {9.0 \;}}}\\ & \underline{-5\;}\\ & \quad 40\\ & \ \underline{-40}\\ & \quad \ \ 0 \end{align*} The decimal form of 9:5 is 1.8.
What about comparing? Can we use decimals to compare ratios?
Sometimes, you may want to compare two ratios and determine if they are equivalent or not. Rewriting both ratios in decimal form is one way to do this.
Example
Compare these two ratios and determine if they are equivalent \begin{align*}\frac{7}{14}\end{align*} and \begin{align*}\frac{11}{20}\end{align*}.
Rewrite \begin{align*}\frac{7}{14}\end{align*} in decimal form.
\begin{align*}& \overset{ \ \ 0.5}{14 \overline{ ) {\ 7.0 \;}}}\\ & \ \ \underline{-70\;}\\ & \qquad 0\\ & \quad \ \underline{-0}\\ & \qquad 0 \end{align*} Rewrite \begin{align*}\frac{11}{20}\end{align*} in decimal form.
\begin{align*}& \overset{ \ \ 0.55}{20 \overline{ ) { 11.00 \;}}}\\ & \ \ \underline{-100\;}\\ & \qquad 100\\ & \quad \ \underline{-100}\\ & \qquad \quad 0 \end{align*} To compare the ratios in decimal form, give each decimal the same number of decimal places. In other words, give 0.5 two decimal places: 0.5 = 0.50.
Now compare. Since both decimals have a 0 in the ones place and a 5 in the tenths place, compare the digits in the hundredths place.
\begin{align*}&0.5\underline{0}\\ &0.5\underline{5}\end{align*}
Since \begin{align*}0 < 5, 0.50 < 0.55\end{align*}. So, the ratios, \begin{align*}\frac{7}{14}\end{align*} and \begin{align*}\frac{11}{20}\end{align*}, are not equivalent.
In fact, \begin{align*}\frac{7}{14} < \frac{11}{20}\end{align*}.
5D. Lesson Exercises
Write each ratio as a decimal.
- 5 to 10
- 4 to 10
- Compare 6 to 10 and 1 to 4
Take a few minutes to check your work with a partner.
IV. Solve Real-World Problems Involving Ratios Among Rational Numbers
Being able to rewrite ratios in different forms can help us solve real-world problems. Let’s look at using ratios to solve some real-world problems.
Example
Elena and Jake have a box with only two colors of marbles in it. There are 28 blue marbles and 16 gray marbles in the box. Elena says that the ratio of gray marbles to blue marbles is \begin{align*}\frac{16}{28}\end{align*}. Jake says that the ratio of gray marbles to blue marbles is \begin{align*}\frac{4}{7}\end{align*}. Who is correct, or are they both correct? They also want to find the ratio of gray marbles to the total number of marbles in the box.
Consider the first question. It is a question about comparing and determining equivalence.
Write the ratio of gray marbles to blue marbles as a fraction. Be careful that the ratio you write compares gray marbles to blue marbles, not blue marbles to gray marbles.
Since there are 16 gray marbles and 28 blue marbles, the ratio is:
\begin{align*}\frac{gray}{blue} = \frac{16}{28}\end{align*}
So, Elena is correct that the ratio of gray marbles to blue marbles is \begin{align*}\frac{16}{28}\end{align*}.
If that ratio is equivalent to \begin{align*}\frac{4}{7}\end{align*}, then Jake is correct too. One way to determine if those two ratios are equivalent is to cross multiply.
\begin{align*}\frac{16}{28} & \overset{?}{=} \frac{4}{7}\\ 28 \times 4 & \overset{?}{=} 16 \times 7\\ 112 & \overset{?}{=} 112\\ 112 & = 112\end{align*}
Since the cross products are equal, the two ratios are equivalent.
So, the answer to the first question is that Elena and Jake are both correct.
Now let’s think about the second part of the question. To figure this out, we need to figure out the ratio of gray marbles to total marbles in the box.
\begin{align*}\frac{gray}{total} = \frac{gray}{gray + blue} = \frac{16}{16 + 28} = \frac{16}{44}\end{align*}
Write that ratio in simplest form.
The factors of 16 are: 1, 2, 4, 8, and 16.
The factors of 44 are: 1, 2, 4, 11, 22, and 44.
The greatest common factor of 16 and 24 is 4. So, divide both terms by 4.
\begin{align*}\frac{16}{44} = \frac{16 \div 4}{44 \div 4} = \frac{4}{11}\end{align*}
The ratio of gray marbles to the total number of marbles, written in simplest form, is \begin{align*}\frac{4}{11}\end{align*}.
Now let’s look at how to apply all that we have learned to the introduction problem.
Real Life Example Completed
Rapid Reading
Now it is time to apply what you have learned in this lesson to our original problem. Here it is again. Reread it and underline any important information.
Kayla and Torrey are both in Mrs. Henderson’s seventh grade Language Arts class. On the first day of school, Mrs. Henderson posed a reading challenge to the class. The challenge was to see how many books they can each read throughout the entire year. The books would be read in silent reading or outside of school and the honor system would be employed. Each student was to keep track of the title and author of each book as they read it. At the end of the year, they will celebrate their accomplishments with a pizza party and students can earn extra credit towards their final grade. While some students were hesitant to take on the challenge, all loved the promise of a pizza party and extra credit. Mrs. Henderson won them over.
Kayla and Torrey have been participating in the challenge since day one. By October, eight weeks into school, Kayla had already finished 6 books.
“I have six read,” Kayla said biting into her sandwich at lunch.
“I think I am reading at the same pace,” Torrey said. “After four weeks, I had already finished 3 books.”
“Are you sure,” Kayla asked.
“Well, I need to count, but I am pretty sure that we have both finished reading the same number of books.”
Is Torrey correct? Have the girls each read the same number of books now that 8 weeks have passed?
To solve this problem, we need to figure out if the girls have been reading the same number of books in the same number of weeks. First, let’s write a ratio that compares Kayla’s books read to the number of weeks she read them in.
Kayla read 6 books in 8 weeks.
\begin{align*}\frac{books}{weeks} = \frac{6}{8}\end{align*}
Torrey said that she believes that she is reading at the same pace as Kayla. She says that she read 3 books in four weeks.
\begin{align*}\frac{books}{weeks} = \frac{3}{4}\end{align*}
We can compare these two ratios to see if they are equivalent and find out if Torrey is reading at the same pace as Kayla. If they are, then the girls have read the same number of books.
To do this, we simplify Kayla’s ratio.
\begin{align*}\frac{6}{8} = \frac{3}{4}\end{align*}
Simplifying Kayla’s ratio shows that she also read 3 books in the first 4 weeks. If Torrey is still reading at this pace, then the girls have both read the same number of books, because both ratios are equivalent.
Vocabulary
- Ratio
- a comparison between two quantities. Ratios can be written as a fraction, with a colon or by using the word to.
- Terms
- the two quantities in a ratio.
- Equivalent Ratios
- when two ratios are equal.
- Simplify
- to write in a simpler form by using the greatest common factor to divide the numerator and the denominator of a fraction by the same number.
- Greatest Common Factor
- the largest common factor between two numbers.
Technology Integration
Khan Academy Ratios as Fractions in Simplest Form
James Sousa, Write a Ratio as a Simplified Fraction
James Sousa, Example of How to Write a Ratio as a Simplified Fraction
Other Videos:
- http://www.mathplayground.com/howto_ratios.html – This video is about understanding the basics of ratios.
- http://www.mathplayground.com/howto_equalratios.html – This video is about understanding equal ratios.
- http://www.mathplayground.com/howto_ratiowordproblems.html – This video is about solving ratio word problems.
Time to Practice
Directions: Determine whether each of the following ratio pairs is equal. Write yes if they are equal and no if they are not equal.
1. \begin{align*}\frac{1}{2}\end{align*} and \begin{align*}\frac{6}{12}\end{align*}
2. \begin{align*}\frac{3}{8}\end{align*} and \begin{align*}\frac{1}{4}\end{align*}
3. \begin{align*}\frac{6}{7}\end{align*} and \begin{align*}\frac{2}{3}\end{align*}
4. \begin{align*}\frac{6}{7}\end{align*} and \begin{align*}\frac{12}{14}\end{align*}
5. \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{10}{15}\end{align*}
6. \begin{align*}\frac{17}{21}\end{align*} and \begin{align*}\frac{6}{7}\end{align*}
7. \begin{align*}\frac{24}{48}\end{align*} and \begin{align*}\frac{12}{24}\end{align*}
8. \begin{align*}\frac{16}{18}\end{align*} and \begin{align*}\frac{32}{38}\end{align*}
9. \begin{align*}\frac{9}{45}\end{align*} and \begin{align*}\frac{1}{9}\end{align*}
10. \begin{align*}\frac{4}{6}\end{align*} and \begin{align*}\frac{44}{66}\end{align*}
Directions: Simplify each ratio. Write the simplified version in the same form as the original ratio.
11. 3 to 6
12. 5:20
13. 18 to 22
14. \begin{align*}\frac{18}{20}\end{align*}
15. \begin{align*}\frac{25}{55}\end{align*}
16. 6 to 42
17. 18 to 10
18. 12 to 4
19. 16:8
20. 24 to 16
Directions: Write each ratio as a decimal.
21. 1 to 4
22. 3 to 6
23. 3:4
24. 8 to 5
25. 7 to 28
Directions: Compare the following ratios using <, > or =.
26. .55 ____1 to 2
27. 3:8 _____ 4 to 9
28. 1 to 2 _____ 4:8