5.13: Fundamentals 12 - Ratios and Proportions
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- 148800
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By the end of this lesson, you should understand
- understand the difference between a ratio and a proportion
- a ratio compares two quantities.
- a ratio can be written in different ways and still be equal to the same value.
- a ratio with different units represents a rate.
- a proportion shows that two ratios are equal.
- there are different ways to set up a proportion.
- you can use different operations to compare fractions or determine the value of a missing quantity.
By the end of this lesson, you should be able to
- recognize when two ratios are equivalent to each other.
- create two equivalent ratios.
- use various techniques, including cross multiplication, to find a missing quantity.
- include units in your ratio or final answer.
FUNDAMENTALS OF THE LESSON
Imagine in a bag of M&M’s there are 2 red M&M's and 5 blue M&M's. We say that the ratio of red M&M's to blue M&M's is 2 to 5. In other words, in this bag for every 2 red M&M’s, we have 5 blue M&M’s. We can write this ratio as 2 : 5.
Now, imagine you have a second bag with 8 red M&M’s and 20 blue M&M’s. In this bag, the ratio of red M&M’s to blue M&M’s is 8 : 20.
You can divide the red M&M’s into 4 groups of 2 and the blue M&M’s into 4 groups of 5. This shows that in this second bag you also have 2 red M&M’s for every 5 blue M&M’s.
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The ratio of red M&M’s to blue M&M’s for the second bag is the same as the first bag. We say that the ratio of 2 : 5 is equivalent to 8 : 20. You can obtain the second ratio by multiplying both numbers of the first ratio by 4. You can obtain the first ratio from the second ratio by dividing both numbers by 4. These are the operations we use to get equivalent fractions. That is why ratios are also written as fractions.
For example, we can write the ratio of red M&M’s to blue M&M’s in the second bag as 8 : 20. Or, we can write it as an equivalent fraction. Since \(\dfrac{8}{20} = \dfrac{8\div4}{20\div4} = \dfrac{2}{5}\), we can express the ratio in lowest terms as \(\dfrac{2}{5}\).
- Set up a ratio to compare 12 feet to 48 feet. Express this ratio in reduced form.
- Express the ratio of 220 miles to 4 hours in reduced form.
- At the local grocery store, eggs cost $2.40 per dozen. What is the cost of 84 eggs?
- An orange drink calls for 2 cans of orange concentrate for every 3 cans of soda. We could say that the ratio of cans to soda is 2 : 3. Complete the table below. Compare the cans of orange concentrate to the cans of soda. What do you notice?
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Cans of Orange Concentrate |
2 |
4 |
6 |
8 |
10 |
12 |
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Cans of Soda |
3 |
6 |
9 |
NEXT STEPS
If the World Were 100 People
Imagine the world only had 100 people… Watch the video entitled “If the World Were 100 People” https://www.youtube.com/watch?v=QFrqTFRy-LU
From the video “If the World were 100 People”, there were 12 people who spoke Mandarin, 6 spoke Spanish, 5 spoke English, 3 spoke Hindu, 3 spoke Arabic, and the rest spoke 6,500 other languages. Use this information for questions 5 and 6.
- How many people speak the 6,500 other languages?
- Express a ratio for:
A. Spanish speaking to the number of people in the world.
B. English speaking to those that speak languages other than English.
C. From the video “If the World Were 100 People”, 21 people were overweight, 63 were at a healthy weight, 15 were malnourished, and 1 person was starving. Find the ratio of malnourished people to those at a healthy weight.
D. What if the world population doubled? If everything else doubled as well, what would the new ratio be for malnourished people to those at a healthy weight? How does this ratio compare to your answer from part (a)?
Which Is Better?
- You are trying to determine the better deal for buying soda. A local supermarket offers a pack of 12 sodas for $3.48. The local discount warehouse offers a pack of 36 sodas for $11.52.
A. Express each deal as a ratio.
B. Which is the better deal?
- A car travels about 26 miles on 1 gallon of gas while a truck travels about 250 miles on 14 gallons of gas. Which gets the better gas mileage?
Making Sure You Have Enough
Sometimes, recipes need to be adjusted for different serving sizes or batches. A bread recipe calls for 5 tablespoons of sugar for every 8 cups of flour.
- Express this comparison of sugar to flour as a ratio.
Solving for an Unknown in a Proportion
Many real-world situations involve the use of proportions. A proportion is an equation showing the equality of two ratios. When we are solving for an unknown in a proportion, we can use a process called cross multiplication. Consider the example below.
Example: A bread recipe calls for 5 tablespoons of sugar for every 8 cups of flour. How many tablespoons of sugar are required for 20 cups of flour?
Solution: Let’s use the variable x to represent the unknown number of tablespoons of sugar in the new ratio. The proportion is shown below on the left. The proportion shown below on the right does not include the units.
We can solve for x by cross multiplying. We multiply diagonally to create an equivalent equation. This is shown by the green arrows below.
The result is that we need 12.5 tablespoons of sugar for 20 cups of flour.
- Let’s find out how many tablespoons of sugar are required for 4 cups of flour.
A. Create a proportion involving the unknown quantity to solve this problem. Use the variable x to represent the unknown quantity. Do not place the units in the proportion.
B. Solve the proportion for x using cross multiplication. Write the answer below.
- Returning to our bread recipe, 5 tablespoons of sugar are needed for every 8 cups of flour. Let’s now find out how many tablespoons of sugar are needed for 32 cups of flour.
A. Create a proportion involving the unknown quantity to solve this problem. Use the variable x to represent the unknown quantity. Do not place the units in the proportion.
B. Solve the proportion for x using cross multiplication. Write in the answer below.
FURTHER APPLICATIONS
- Kiki, Tre, and Julie made $2,520 for working at a tree nursery, but they did not work the same amount of hours. Kiki worked 30 hours, Tre worked 50 hours, and Julie worked 60 hours. They divided the money in proportion to the number of hours worked. If they all earn the same rate of pay, how much money did each person earn?
- A fire truck can hold 3000 gallons of water. A firefighter can deliver 160 gallons of water every 2 minutes from the truck.
A. How much water will be delivered in 10 minutes?
B. How long will it take for the firefighter to empty the tank?
Questions: Ratios and Proportions
1. Write two ratios that are equal to \(\dfrac{1}{4}\).
2. Write each as a ratio in lowest terms:
A. 15 miles to 20 miles
B. 136 hours to 20 days
3. Fill in the missing value to make an equivalent statement.
A. \(\dfrac{3}{2} =\dfrac{?}{18}\)
B. \(\dfrac{4}{5} = \dfrac{12}{?}\)
- Determine if each of the following statements is a proportion:
A. \(\dfrac{3}{5} = \dfrac{6}{9}\)
B. \(\dfrac{10}{9} =\dfrac{29}{27}\)
C. \(\dfrac{5}{7} = \dfrac{15}{21}\)
D. \(\dfrac{8}{9} = \dfrac{16}{27}\)
- Solve each proportion:
A. \(\dfrac{a}{8} = \dfrac{5}{2}\)
B. \(\dfrac{20}{15} = \dfrac{x}{14}\) (Round to 2 decimal places.)
6. A pancake recipe that feeds three people uses 2 cups of flour and 1 cup of milk.
A. Express this as a ratio with units (flour to milk).
B. Create an equivalent ratio that would feed 12 people.
- There are 20 pieces of fruit in a bowl. Three pieces are bananas. Use this information to express the ratio of:
A. Bananas to total fruit.
B. Total fruit to bananas.
C. Bananas to fruit other than bananas.
- Michael takes 1 hour to type 5 pages of an essay. Determine the number of hours he would need to type 27 pages. Create and use a proportion to solve this problem to the nearest tenth.
- A 16-oz grande Shaken Sweet Tea at Starbucks has 100 calories.15 How many calories are there in a 24-oz venti Shaken Sweet Tea? Create and use a proportion to solve this problem.
- If a 170-pound person weighs approximately 65 pounds on Mars, how much does a 9000-pound satellite weigh on Mars? Create and use a proportion to solve this problem. Round to the nearest pound.
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15 Starbucks, Shaken Sweet Tea, accessed August 1, 2018, https://www.starbucks.com/menu/drink...?foodZone=9999