We can use ratios to solve problems by setting up a proportion. A propor- tion is a relationship between two equivalent ratios. A proportion usually contains a reduced ratio that represents a relationship between two quan- tities, and a larger, equivalent ratio that represents the exact values of those two quantities. The ratio 16:12 is equivalent to the ratio 4:3. These equiva- lent ratios can be expressed as a proportion: 1162= 43.
If the ratio of quantity A to quantity B is 7:2, and the exact value of quan- tity A is 35, we can set up a proportion to solve for x, the exact value of quan- tity B. As a fraction, 7:2 is 27. Set that fraction equal to the exact value of
quantity A over the exact value of quantity B: 27= 3x5. Cross multiply and set the products equal to each other: 7x = 70, x = 10. If the exact value of quan- tity A is 35, then exact value of quantity B is 10.
If the ratio of quantity A to quantity B is 7:2, and the exact total of quan- tity A and quantity B is 27, we can find the exact value of quantity A and quan- tity B by writing ratios that represent the part-to-whole relationship. If the ratio of quantity A to quantity B is 7:2, then 7 of every 7 + 2 = 9 items are of quantity A. The ratio of quantity A to the whole is 7:9. The whole is 27, and exact number of quantity A is x: 97= 2x7. Cross multiply and set the prod- ucts equal to each other: 9x = 189, x = 21. If the total of quantity A and quan- tity B is 27, then the total of quantity A is 21. The total of quantity B can be found with the proportion 92= 2x7. 9x = 54, x = 6. If the total of quantity A and quantity B is 27, then the total of quantity B is 6.
Now that we remember how to use ratios and proportions, let’s look at how to recognize and solve ratio and proportion word problems.
Ratio word problems often contain the word ratio or every. You may also see the ratio expressed in colon form, such as 2:3. Once you’ve recognized a word problem as a ratio and proportion problem, decide if it is a part-to- part problem or a part-to-whole problem. A part-to-part problem gives you a ratio and the value of one quantity, and asks you for the value of the other quantity. A part-to-whole problem gives you a ratio and the total value of both quantities, and asks you for the value of one quantity.
INSIDE TRACK
ALTHOUGH THE KEYWORDtotal can often signal addition, in a
ratio problem, the word total can signal a part-to-whole problem rather than a part-to-part problem.
Example
The ratio of peanuts to cashews in a jar of nuts is 5:6, and the exact number of peanuts in the jar is 25. How many cashews are in the jar?
Read the entire word problem.
We are given the ratio of peanuts to cashews and the exact number of peanuts.
Identify the question being asked.
We are looking for the number of cashews.
Underline the keywords and words that indicate formulas.
The word ratio indicates that we will either be finding a ratio or using a ratio to set up a proportion.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
This problem contains two quantities, of peanuts and of cashews. We are given the value of one quantity, and we are asked for the value of the other quantity. This is a part-to-part problem. We must set up a proportion that compares the ratio of peanuts to cashews to the actual numbers of peanuts and cashews.
Write number sentences for each operation.
The ratio of peanuts to cashews is 5:6, or 65. Use x to represent the actual number of cashews, since that number is unknown. The ratio of actual peanuts to cashews is 25:x, or 2x5. Set these fractions equal to each other:
65= 2x5
Solve the number sentences and decide which answer is reasonable.
Cross multiply and solve for x:
(6)(25) = 5x, 150 = 5x, x = 30. If there are 25 peanuts in the jar, then there must be 30 cashews in the jar.
Check your work.
The ratio of peanuts to cashews is 5:6, so we can check our answer by comparing that ratio to the actual numbers of peanuts and cashews. The actual numbers of peanuts and cashews should reduce to the ratio 5:6. The greatest common factor of 25 and 30 is 5, and 25:30 does reduce to 5:6. Our answer is correct.
INSIDE TRACK
BEFORE SETTING UPa proportion, be sure any given ratios are in reduced form. This will make cross multiplying and solving the pro- portion easier.
Example
Dermot finds that the ratio of basketball players to football players at his school is 3:15. If there are 90 total players, how many of the players are basketball players?
Read the entire word problem.
We are given the ratio of basketball players to football players and the total number of players.
Identify the question being asked.
We are looking for the number of basketball players.
Underline the keywords and words that indicate formulas.
The word ratio indicates that we will either be finding a ratio or using a ratio to set up a proportion. The word total is a signal that this is a part-to-whole ratio problem.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
This problem contains two quantities of basketball players and of foot- ball players. We are given the total of the quantities and we are asked for the value of one quantity. This is a part-to-whole problem. Since we are looking for the number of basketball players, we must set up a proportion that compares the ratio of basketball players to total play- ers to the actual numbers of basketball players and total players.
Write number sentences for each operation.
The ratio of basketball players to football players is 3:15, which means that the ratio of basketball players to total players is 3:(15 + 3), which is 3:18. Reduce this ratio by dividing both numbers by 3: 3:18 = 1:6, or 61. Use x to represent the actual number of basketball
players, since that number is unknown. There are 90 total players, so the ratio of actual basketball players to total players is x:90, or 9x0. Set these fractions equal to each other:
61= 9x0
Solve the number sentences and decide which answer is reasonable.
Cross multiply and solve for x:
6x = 90, x = 15. If there are 90 total players, then there must be 15 basketball players.
Check your work.
The ratio of basketball players to total players is 1:6. Check that the actual number of players and the total number of players are in the ratio 1:6; 15:90 = 1:6, so our answer is correct.
INSIDE TRACK IN THE LASTexample, we used a ratio and the fact that there are 90 total players to find that there are 15 basketball players. What if we wanted to find the number of football players? We could repeat the same process, or we could subtract the number of basketball players from the total number of players: 90 – 15 = 75 football play- ers. Given a total, once we find the actual value of one quantity, we can use subtraction to find the value of the other quantity.
We’ve used proportions and the value of one quantity to find the value of another quantity, and we’ve used proportions and the total of two quantities to find the value of one quantity. We can also use proportions and the value of one quantity to find the total of two quantities.
Example
Caroline’s summer camp has indoor days and outdoor days, depend- ing on the weather. The ratio of indoor days to outdoor days was 3:17. If the camp had 34 outdoor days, for how many total days did Caroline attend camp?
Read the entire word problem.
We are given the ratio of indoor days to outdoor days and the actual number of outdoor days.
Identify the question being asked.
We are looking for the total number of days that Caroline attended camp.
Underline the keywords and words that indicate formulas.
The word ratio indicates that we will either be finding a ratio or using a ratio to set up a proportion. The word total is a signal that this is a part-to-whole ratio problem.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Again, this problem contains two quantities, indoor days and outdoor days. We are given the value of one quantity, outdoor days, and we are asked for the total value. This is a part-to-whole problem. Since we are looking for the total number of days, we must set up a proportion that compares the ratio of outdoor days to total days to the actual numbers of outdoor days and total days.
Write number sentences for each operation.
The ratio of indoor days to outdoor days is 3:17, which means that the ratio of outdoor days to total days is 17:(17 + 3), which is 17:20, or 2170. Use x to represent the total number of days, since that number is unknown. There are 34 outdoor days, so the ratio of outdoor days to total days is 34:x, or 3x4. Set these fractions equal to each other:
2170= 3x4
Solve the number sentences and decide which answer is reasonable.
Cross multiply and solve for x:
17x = 680, x = 40. If there are 34 outdoor days, then there must be 40 total days.
Check your work.
The ratio of outdoor days to total days is 17:20. Check that the actual number of outdoor days and the total number of days are in the ratio 17:20; 34:40 = 17:20, so our answer is correct.
PRACTICE L AP
6.The ratio of chairs to tables in the school cafeteria is 8:1. If there are 96 chairs in the cafeteria, how many tables are in the cafeteria?
7. The ratio of orange jelly beans to lemon jelly beans in a jar is 4:9. If there are 99 lemon jelly beans in the jar, how many orange jelly beans are in the jar?
8.There are 56 volunteers at the community center. If the ratio of paid employees to volunteers is 3:7, how many paid employees are at the community center?
9.Nick collects books. The ratio of his fiction books to his nonfic- tion books is 11:6. If Nick has 154 fiction books, how many non- fiction books does he have?
10.A card store sold 165 birthday cards and Valentine’s Day cards in total on Sunday. If the ratio of birthday cards to Valentine’s Day cards is 2:9, how many birthday cards did the store sell?
11.A rain forest contains many animals, including orangutans and gorillas. The ratio of orangutans to gorillas is 5:3. If the total num- ber of orangutans and gorillas is 312, how many gorillas are in the rain forest?
12.A total of 88 students write for either the school newspaper or the yearbook. If the ratio of yearbook writers to newspaper writ- ers is 13:9, how many students write for the yearbook?
13.A train conductor sells one-way tickets and round-trip tickets. The ratio of one-way tickets sold to round-trip tickets sold is 8:15. If the conductor sold 75 round-trip tickets, how many total tick- ets did he sell?
14.A magazine company hires part-time employees and full-time employees. The ratio of part-time employees to full-time employ- ees is 5:14, and 168 of them are full-time employees. How many total people are employed at the magazine?
15.The ratio of Scottsdale residents to Wharton residents is 7:12. If 8,435 people live in Scottsdale, how many people live in Scottsdale and Wharton combined?
PACE YOURSELF
OF YOUR FRIENDS,what is the ratio of boys to girls? Look up the population of your town. If the ratio of men to women in your town was the same as the ratio of boys to girls of your friends, how many men and how many women would there be in your town?
SUMMARY
THE WORD RATIOor the symbol “:” usually indicates that we will need a proportion to solve a word problem. Once we know that we are working on a ratio word problem, the word total can help us determine if it is a part-to-part problem or a part-to-whole problem.
ANSWERS
1. If there are five adults and 25 students, then the ratio of adults to stu- dents is 5 to 25, or 5:25. We can reduce this ratio by dividing both num- bers by 5; 5:25 = 1:5.
2. If there are four roses and 12 daffodils, then the ratio of roses to daffodils is 4:12 and the ratio of daffodils to roses is 12:4. We can reduce this ratio by dividing both numbers by 3; 12:4 = 3:1.
3. If there are 28 oak trees and 44 total trees, and every tree is an oak or a maple, then there are 44 – 28 = 16 maple trees. The ratio of oak trees to maple trees is 28:16. We can reduce this ratio by dividing both num- bers by 4; 28:16 = 7:4.
4. Ellie’s purse contains 15 + 10 = 25 total coins. Since 15 of them are nick- els, the ratio of nickels to total coins is 15:25. We can reduce this ratio by dividing both numbers by 5; 15:25 = 3:5.
5. The box contains 8 + 4 = 12 total doughnuts. Since four of them are glazed, the ratio of glazed doughnuts to total doughnuts is 4:12. We can reduce this ratio by dividing both numbers by 4; 4:12 = 1:3.