Have you ever participated in a reading contest or challenge?
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 is 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 is to keep track of the title and author of each book as they read it. At the end of the year, they would 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 Concept, pay attention and you be able to figure out the book dilemma at the end of the Concept.
Guidance
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 to us as people. 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. For example, 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 total apples in the basket as a fraction, too.
\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.
Now that you know how to write a ratio, let’s think about 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.
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 they simplify to the same numbers, 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 both 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.
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.
Determine whether or not each pair of ratios is equivalent. Write yes or no.
Example A
\begin{align*}\frac{3}{4}\end{align*} and \begin{align*}\frac{9}{12}\end{align*}
Solution: Yes
Example B
\begin{align*}\frac{5}{6}\end{align*} and \begin{align*}\frac{20}{30}\end{align*}
Solution: No
Example C
\begin{align*}\frac{4}{5}\end{align*} and \begin{align*}\frac{8}{10}\end{align*}
Solution: Yes
Now it is time to apply what you have learned in this lesson to our original problem. Here it is again.
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 is 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 is to keep track of the title and author of each book as they read it. At the end of the year, they would 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.
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 on the same pace as Kayla. She says that she read 3 books after four weeks.
\begin{align*}\frac{books}{weeks} = \frac{3}{4}\end{align*}
If Torrey is reading at the same pace, then we can compare these two ratios to see if they are equivalent. If they are, then the girls have been reading 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.
Guided Practice
Here is one for you to try on your own.
Determine if these two ratios are equivalent \begin{align*}\frac{4}{8}\end{align*} and \begin{align*}\frac{1}{2}\end{align*}.
Answer
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.
Video Review
- This is a James Sousa video on ratios. This video on ratios supports success in this Concept.
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*}
11. \begin{align*}\frac{6}{9}\end{align*} and \begin{align*}\frac{4}{6}\end{align*}
12. \begin{align*}\frac{14}{16}\end{align*} and \begin{align*}\frac{20}{24}\end{align*}
13. \begin{align*}\frac{12}{16}\end{align*} and \begin{align*}\frac{24}{32}\end{align*}
14. \begin{align*}\frac{24}{48}\end{align*} and \begin{align*}\frac{1}{2}\end{align*}
15. \begin{align*}\frac{84}{96}\end{align*} and \begin{align*}\frac{3}{4}\end{align*}