Find practice problems and solutions for these concepts at Ratios and Proportions in Word Problems Practice Problems.
We can use relationships between numbers or pairs of numbers to solve problems. For instance, let's say a deck of cards contains twice as many red cards as black cards. If we know the number of red cards, we can find the number of black cards, because we know the relationship between the number of red cards and the number of black cards. This kind of relationship is called a ratio. A ratio is a comparison, or relationship, between two numbers. Ratios can be shown using a colon or as a fraction. The ratio 3 to 2 can be written as 3:2 or .
Ratios: Subject Review
If there are 16 black pens and 12 blue pens on a desk, then the ratio of black pens to blue pens is 16:12. Ratio can be reduced just like fractions, since ratios are fractions. The greatest common factor of 16 and 12 is 4, so both numbers can be divided by 4. The ratio 16:12 is the same as the ratio 4:3.
Caution
The order of the numbers in a ratio is important. The ratio 3:4 is not the same as the ratio 4:3. Remember, ratios are fractions, and the fraction is not equal to the fraction . If you are describing the ratio of apples to oranges, place the number of apples first in your ratio.

If a word problem simply asks you to find or write a ratio, you may not need the eightstep process to find the answer. However, as with any kind of word problem, the eightstep process will work, and you should use it if you find that you cannot solve a word problem quickly.
Example
If there are 20 red cars and 24 blue cars in a parking lot, what is the ratio of red cars to blue cars?
This problem asks us to find a ratio, and it gives us the number of red cars and the number of blue cars. No computation is needed. The ratio of red cars to blue cars is 20:24. We can reduce the ratio to 5:6.
Example
A parking lot contains only red cards and blue cars. If there are 36 total cars and 16 of them are red, what is the ratio of blue cars to red cars?
Again, we are asked to write a ratio, but this time, we are not given the number of blue cars. We can find the number of blue cars by subtracting the number of red cars from the total number of cars: 36 – 16 = 20. We must be sure to write our ratio in the correct order. The problem asks us to find the ratio of blue cars to red cars, which is 20:16, or 5:4. Determining the question being asked can make the difference between a right or wrong answer.
Pace Yourself
Find the ratio of $1 bills to $5 bills in your wallet or purse. What would that ratio be if you added three $5 bills to your wallet or purse?

Proportions: Subject Review
We can use ratios to solve problems by setting up a proportion. A proportion is a relationship between two equivalent ratios. A proportion usually contains a reduced ratio that represents a relationship between two quantities, 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 equivalent ratios can be expressed as a proportion: .
If the ratio of quantity A to quantity B is 7:2, and the exact value of quantity A is 35, we can set up a proportion to solve for x, the exact value of quantity B. As a fraction, 7:2 is . Set that fraction equal to the exact value of quantity A over the exact value of quantity B: . Cross multiply and set the products equal to each other: 7x = 70, x = 10. If the exact value of quantity 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 quantity A and quantity B is 27, we can find the exact value of quantity A and quantity B by writing ratios that represent the parttowhole 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: . Cross multiply and set the products equal to each other: 9x = 189, x = 21. If the total of quantity A and quantity B is 27, then the total of quantity A is 21. The total of quantity B can be found with the proportion . 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 parttopart problem or a parttowhole problem. A parttopart problem gives you a ratio and the value of one quantity, and asks you for the value of the other quantity. A parttowhole 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 keyword total can often signal addition, in a ratio problem, the word total can signal a parttowhole problem rather than a parttopart 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 parttopart 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 . 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 . Set these fractions equal to each other:
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 up a proportion, be sure any given ratios are in reduced form. This will make cross multiplying and solving the proportion 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 parttowhole 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 football players. We are given the total of the quantities and we are asked for the value of one quantity. This is a parttowhole 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 players 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 . 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 . Set these fractions equal to each other:
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 last example, 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 players. 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, depending 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 parttowhole 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 parttowhole 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 . 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 . Set these fractions equal to each other:
=
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.
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 ratio or 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 parttopart problem or a parttowhole problem.
Find practice problems and solutions for these concepts at Ratios and Proportions in Word Problems Practice Problems.