Lesson 11
Percentages and Double Number Lines
Let’s use double number lines to represent percentages.
11.1: Fundraising Goal
Each of three friends—Lin, Jada, and Andre—had the goal of raising $40. How much money did each person raise? Be prepared to explain your reasoning.
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Lin raised 100% of her goal.
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Jada raised 50% of her goal.
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Andre raised 150% of his goal.
11.2: Three-Day Biking Trip
Elena biked 8 miles on Saturday. Use the double number line to answer the questions. Be prepared to explain your reasoning.
![A double number line.](https://staging-cms-im.s3.amazonaws.com/9gJiUP1Mq7oHfFc8p5JvjL31?response-content-disposition=inline%3B%20filename%3D%226-6.3.D2_Image_2.png%22%3B%20filename%2A%3DUTF-8%27%276-6.3.D2_Image_2.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093720Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e9f1f8156e72f238522a9beb8f15623c140df0e5c21df78b3fdc76090a7c2f98)
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What is 100% of her Saturday distance?
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On Sunday, she biked 75% of her Saturday distance. How far was that?
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On Monday, she biked 125% of her Saturday distance. How far was that?
11.3: Puppies Grow Up
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Jada has a new puppy that weighs 9 pounds. The vet says that the puppy is now at about 20% of its adult weight. What will be the adult weight of the puppy?
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Andre also has a puppy that weighs 9 pounds. The vet says that this puppy is now at about 30% of its adult weight. What will be the adult weight of Andre’s puppy?
- What is the same about Jada and Andre’s puppies? What is different?
A loaf of bread costs $2.50 today. The same size loaf cost 20 cents in 1955.
- What percentage of today’s price did someone in 1955 pay for bread?
- A job pays $10.00 an hour today. If the same percentage applies to income as well, how much would that job have paid in 1955?
Summary
We can use a double number line to solve problems about percentages. For example, what is 30% of 50 pounds? We can draw a double number line like this:
![A double number line. Weight, pounds.](https://staging-cms-im.s3.amazonaws.com/dutV3bkoPWfjvF11wqN7CBKf?response-content-disposition=inline%3B%20filename%3D%226-6.3.D2_Image_15.png%22%3B%20filename%2A%3DUTF-8%27%276-6.3.D2_Image_15.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093721Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6e1e098848ca7b6c94c14025670b2b7846a5f5178cbd595c40da8866904a478a)
We divide the distance between 0% and 100% and that between 0 and 50 pounds into ten equal parts. We label the tick marks on the top line by counting by 5s (\(50 \div 10 = 5\)) and on the bottom line counting by 10% (\(100 \div 10 =10\)). We can then see that 30% of 50 pounds is 15 pounds.
We can also use a table to solve this problem.
![A 2-column table with 3 rows of data. First column, weight, pounds. Second column, percentage.](https://staging-cms-im.s3.amazonaws.com/mYzmETFZBYoFbgu6NU9Cb9fk?response-content-disposition=inline%3B%20filename%3D%226-6.3.D2_Image_16.png%22%3B%20filename%2A%3DUTF-8%27%276-6.3.D2_Image_16.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093721Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b18919704c03611ca2b4ca9f16c9021a06c45db8b6d1a66483a152df8b61eb84)
Suppose we know that 140% of an amount is \$28. What is 100% of that amount? Let’s use a double number line to find out.
![A double number line.](https://staging-cms-im.s3.amazonaws.com/SjWWuAGjacN7NvGWttGqQo2B?response-content-disposition=inline%3B%20filename%3D%226-6.3.D2_Image_17.png%22%3B%20filename%2A%3DUTF-8%27%276-6.3.D2_Image_17.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093721Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5876510098868427bcc74eac90c226fd9aa76d5aa2a8ba54f53972fde11084c9)
We divide the distance between 0% and 140% and that between \$0 and \$28 into fourteen equal intervals. We label the tick marks on the top line by counting by 2s and on the bottom line counting by 10%. We would then see that 100% is \$20.
Or we can use a table as shown.
![A two column table with three rows of data.](https://staging-cms-im.s3.amazonaws.com/5PAyofUSUdYuL6kA7kVnB7PR?response-content-disposition=inline%3B%20filename%3D%226-6.3.D2_Image_22.png%22%3B%20filename%2A%3DUTF-8%27%276-6.3.D2_Image_22.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093721Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5dd59a15be1387af51364f6a9808c8df973f1c85abd6e06ee10c64a29d36976f)
Glossary Entries
- percent
The word percent means “for each 100.” The symbol for percent is %.
For example, a quarter is worth 25 cents, and a dollar is worth 100 cents. We can say that a quarter is worth 25% of a dollar.
- percentage
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.