To review these concepts, go to Ratios and Proportions in Word Problems Study Guide.

### Practice

**Directions:** Answer the following questions and reduce all ratios.

- If there are five adults and 25 students on a field trip, what is the ratio of adults to students?
- If there are four roses and 12 daffodils in a vase, what is the ratio of daffodils to roses?
- Every tree in Woodmere Park is either an oak or a maple. If there are 28 oak trees and 44 total trees, what is the ratio of oak trees to maple trees?
- Ellie's purse contains 15 nickels and ten dimes. What is the ratio of nickels to total coins?
- A box contains eight jelly doughnuts and four glazed doughnuts. What is the ratio of glazed doughnuts to total doughnuts?
- 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?
- 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?
- 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?
- Nick collects books. The ratio of his fiction books to his nonfiction books is 11:6. If Nick has 154 fiction books, how many non- fiction books does he have?
- 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?
- A rain forest contains many animals, including orangutans and gorillas. The ratio of orangutans to gorillas is 5:3. If the total number of orangutans and gorillas is 312, how many gorillas are in the rain forest?
- A total of 88 students write for either the school newspaper or the yearbook. If the ratio of yearbook writers to newspaper writers is 13:9, how many students write for the yearbook?
- 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 tickets did he sell?
- A magazine company hires part-time employees and full-time employees. The ratio of part-time employees to full-time employees is 5:14, and 168 of them are full-time employees. How many total people are employed at the magazine?
- 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?

### Solutions

- If there are five adults and 25 students, then the ratio of adults to students is 5 to 25, or 5:25. We can reduce this ratio by dividing both numbers by 5; 5:25 = 1:5.
- 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.
- 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 numbers by 4; 28:16 = 7:4.
- Ellie's purse contains 15 + 10 = 25 total coins. Since 15 of them are nickels, 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.
- 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.
*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.

We are given the ratio of chairs to tables and the exact number of chairs.

*Identify the question being asked*.

We are looking for the number of tables.

*Underline the keywords and words that indicate formulas*.

The word *ratio* means that we will likely be 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, chairs and tables. We are given the value of one quantity, chairs, and we are asked for the value of the other quantity, tables. This is a part-to-part problem. We must set up a proportion that compares the ratio of chairs to tables to the ratio of the actual numbers of chairs to tables.

*Write number sentences for each operation*.

The ratio of chairs to tables is 8:1, or . Use *x* to represent the actual number of tables, since that number is unknown. The ratio of actual chairs to tables is 96:*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*

(96)(1) = 8*x*, 96 = 8*x*, *x* = 12. If there are 96 chairs in the cafeteria, then there must be 12 tables in the cafeteria.

*Check your work*.

The ratio of chairs to tables is 8:1, so we can check our answer by comparing that ratio to the actual numbers of chairs to tables. The actual numbers of chairs and tables should reduce to the ratio 8:1. The greatest common factor of 96 and 12 is 8, and 96:12 does reduce to 8:1.

We are given the ratio of orange jelly beans to lemon jelly beans and the exact number of lemon jelly beans.

*Identify the question being asked*.

We are looking for the number of orange jelly beans.

*Underline the keywords and words that indicate formulas*.

The word *ratio* means that we will likely be 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, orange jelly beans and lemon jelly beans. We are given the value of one quantity, lemon jelly beans, and we are asked for the value of the other quantity, orange jelly beans. This is a part-to-part problem. We must set up a proportion that compares the ratio of orange jelly beans to lemon jelly beans to the ratio of the actual numbers of orange jelly beans to lemon jelly beans.

*Write number sentences for each operation*.

The ratio of orange jelly beans to lemon jelly beans is 4:9, or Use *x* to represent the actual number of orange jelly beans, since that number is unknown. The ratio of actual orange jelly beans to lemon jelly beans is *x*:99, or . Set these fractions equal to each other:

=

*Solve the number sentences and decide which answer is reasonable*.

Cross multiply and solve for *x*:

9*x* = (4)(44), 9 *x* = 396, *x* = 44. If there are 99 lemon jelly beans in the jar, then there must be 44 orange jelly beans in the jar.

*Check your work*.

The ratio of orange jelly beans to lemon jelly beans is 4:9, so the ratio of actual orange jelly beans to actual lemon jelly beans should reduce to 4:9. The greatest common factor of 44 and 99 is 11, and 44:99 reduces to 4:9.

We are given the ratio of paid employees to volunteers and the exact number of volunteers.

*Identify the question being asked*.

We are looking for the number of paid employees.

*Underline the keywords and words that indicate formulas*.

The word *ratio* means that we will likely be 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, paid employees and volunteers. We are given the value of one quantity, volunteers, and we are asked for the value of the other quantity, paid employees. This is a part-to-part problem. We must set up a proportion that compares the ratio of paid employees to volunteers to the ratio of actual paid employees to actual volunteers.

*Write number sentences for each operation*.

The ratio of paid employees to volunteers is 3:7, or . Use *x* to represent the actual number of paid employees, since that number is unknown. The ratio of actual paid employees to actual volunteers is *x*:56, or . Set these fractions equal to each other:

= .

*Solve the number sentences and decide which answer is reasonable*.

Cross multiply and solve for *x*:

7*x* = (3)(56), 7*x* = 168, *x* = 24. If there are 56 volunteers at the community center, then there must be 24 paid employees at the center.

*Check your work*.

The ratio of paid employees to volunteers is 3:7, so the ratio of actual paid employees to actual volunteers should reduce to 3:7. The greatest common factor of 24 and 56 is 8, and 24:56 reduces to 3:7.

We are given the ratio of fiction books to nonfiction books and the exact number of fiction books.

*Identify the question being asked*.

We are looking for the number of nonfiction books.

*Underline the keywords and words that indicate formulas*.

The word *ratio* means that we will likely be 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, fiction books and nonfiction books. We are given the value of one quantity, fiction books, and we are asked for the value of the other quantity, nonfiction books. This is a part-to-part problem. We must set up a proportion that compares the ratio of fiction books to nonfiction books to the ratio of actual fiction books to actual nonfiction books.

*Write number sentences for each operation*.

The ratio of fiction books to nonfiction books is 11:6, or . Use *x* to represent the actual number of nonfiction books, since that number is unknown. The ratio of actual fiction books to actual nonfiction books is 154:*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*:

11*x* = (154)(6), 11*x* = 924, *x* = 84. If Nick has 154 fiction books, then he has 84 nonfiction books.

*Check your work*.

The ratio of fiction books to nonfiction books is 11:6, so the ratio of actual fiction books to actual nonfiction books should reduce to 11:6. The greatest common factor of 154 and 84 is 14, and 154:84 reduces to 11:6.

We are given the ratio of birthday cards and Valentine's Day cards and the total number of cards sold.

*Identify the question being asked*.

We are looking for the number of birthday cards sold.

*Underline the keywords and words that indicate formulas*.

The word ratio means that we will likely be 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, birthday cards and Valentine's Day cards. We are given the total of the two quantities, and we are looking for the value of one quantity. This is a part-to-whole problem. Since we are looking for the number of birthday cards sold, we must set up a proportion that compares the ratio of birthday cards sold to total cards sold to the ratio of actual birthday cards sold to actual total cards sold.

*Write number sentences for each operation*.

The ratio of birthday cards to Valentine's Day cards is 2:9, which means that the ratio of birthday cards to total cards is 2:(2 + 9), which is 2:11, or . Use *x* to represent the number of birthday cards sold, since that number is unknown. There are 165 total cards sold, so the ratio of birthday cards sold to total cards sold is *x*:165, or . Set these fractions equal to each other:

=

*Solve the number sentences and decide which answer is reasonable*.

Cross multiply and solve for *x*:

11*x* = (2)(165), 11*x* = 330, *x* = 30. If there are 165 total cards sold, then 30 of them were birthday cards.

*Check your work*.

The ratio of birthday cards to total cards is 2:11, so the ratio of actual birthday cards sold to total cards sold should reduce to 2:11. The greatest common factor of 30 and 165 is 15, and 30:165 reduces to 2:11.

We are given the ratio of orangutans to gorillas and the total number of orangutans and gorillas.

*Identify the question being asked*.

We are looking for the number of gorillas.

*Underline the keywords and words that indicate formulas*.

The word *ratio* means that we will likely be 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, orangutans and gorillas. We are given the total of the two quantities, and we are looking for the value of one quantity. This is a part-to-whole problem. Since we are looking for the number of gorillas, we must set up a proportion that compares the ratio of gorillas to the total number of orangutans and gorillas to the ratio of actual orangutans to the total number of actual orangutans and gorillas.

*Write number sentences for each operation*.

The ratio of orangutans to gorillas is 5:3, which means that the ratio of gorillas to the total number of orangutans and gorillas is 3:(3 + 5), which is 3:8, or . Use *x* to represent the number of gorillas, since that number is unknown. There are 312 total orangutans and gorillas, so the ratio of gorillas to total orangutans and gorillas is *x*:312, or . Set these fractions equal to each other:

=

*Solve the number sentences and decide which answer is reasonable*.

Cross multiply and solve for *x*:

8*x* = (3)(312), 8*x* = 936, *x* = 117. If there are 312 total cards orangutans and gorillas, then 117 of them are gorillas.

*Check your work*.

The ratio of gorillas to total orangutans and gorillas is 3:8, so the ratio of actual gorillas to actual total orangutans and gorillas should reduce to 3:8. The greatest common factor of 117 and 312 is 39, and 117:312 reduces to 3:8.

We are given the ratio of yearbook writers to newspaper writers and the total number of writers.

*Identify the question being asked*.

We are looking for the number of yearbook writers.

*Underline the keywords and words that indicate formulas*.

The word *ratio* means that we will likely be 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, yearbook writers and newspaper writers. We are given the total of the two quantities, and we are looking for the value of one quantity. This is a part-to-whole problem. Since we are looking for the number of yearbook writers, we must set up a proportion that compares the ratio of yearbook writers to the total number of writers to the ratio of actual yearbook writers to the total number of actual writers.

*Write number sentences for each operation*.

The ratio of yearbook writers to newspaper writers is 13:9, which means that the ratio of yearbook writers to the total number of writers is 13:(9 + 13), which is 13:22, or . Use *x* to represent the number of yearbook writers, since that number is unknown. There are 88 total writers, so the ratio of yearbook writers to total writers is *x*:88, or . Set these fractions equal to each other:

=

*Solve the number sentences and decide which answer is reasonable*.

Cross multiply and solve for *x*:

22*x* = (13)(88), 22*x* = 1,144, *x* = 52. If there are 88 total writers, then 52 of them are yearbook writers.

*Check your work*.

The ratio of yearbook writers to total writers is 13:22, so the ratio of actual yearbook writers to actual total writers should reduce to 13:22. The greatest common factor of 52 and 88 is 4, and 52:88 reduces to 13:22.

We are given the ratio of one-way tickets to round-trip tickets and the number of round-trip tickets.

*Identify the question being asked*.

We are looking for the total number of tickets.

*Underline the keywords and words that indicate formulas*.

*ratio* means that we will likely be 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, one-way tickets and round-trip tickets. We are given the value of one quantity, and we are looking for the total of the two quantities. This is a part-to-whole problem. Since we are looking for the total number of tickets, we must set up a proportion that compares the ratio of round-trip tickets to the total number of tickets to the ratio of actual round-trip tickets to the total number of actual tickets.

*Write number sentences for each operation*.

The ratio of one-way tickets to round-trip tickets is 8:15, which means that the ratio of round-trip tickets to the total number of tickets is 15:(8 + 15), which is 15:23, or . Use *x* to represent the number of total tickets, since that number is unknown. There are 75 round-trip tickets, so the ratio of round-trip tickets to total tickets is 75:*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*:

15*x* = (75)(23), 15*x* = 1,725, *x* = 115. If there were 75 round-trip tickets sold, then there were 115 total tickets sold.

*Check your work*.

The ratio of round-trip tickets to the total number of tickets is 15:23, so the ratio of actual round-trip tickets to the total number of actual tickets should reduce to 15:23. The greatest common factor of 75 and 115 is 5, and 75:115 reduces to 15:23.

We are given the ratio of part-time employees to full-time employees and the number of full-time employees.

*Identify the question being asked*.

We are looking for the total number of employees.

*Underline the keywords and words that indicate formulas*.

*ratio* means that we will likely be 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, part-time employees and full-time employees. We are given the value of one quantity, and we are looking for the total of the two quantities. This is a part-to-whole problem. Since we are looking for the total number of employees, we must set up a proportion that compares the ratio of full-time employees to the total number of employees to the ratio of actual full-time employees to the total number of actual employees.

*Write number sentences for each operation*.

The ratio of part-time employees to full-time employees is 5:14, which means that the ratio of full-time employees to the total number of employees is 14:(5 + 14), which is 14:19, or . Use *x* to represent the number of total tickets, since that number is unknown. There are 168 full-time employees, so the ratio of full-time employees to total employees is 168:*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*:

14*x* = (168)(19), 14*x* = 3,192, *x* = 228. If there are 168 full-time employees, then there are 228 total employees.

*Check your work*.

The ratio of full-time employees to the total number of employees is 14:19, so the ratio of actual full-time employees to the total number of actual employees should reduce to 14:19. The greatest common factor of 168 and 228 is 12, and 168:228 reduces to 14:19.

We are given the ratio of Scottsdale residents to Wharton residents and the number of Scottsdale residents.

*Identify the question being asked*.

We are looking for the total number of residents in the two towns.

*Underline the keywords and words that indicate formulas*.

The word *ratio* means that we will likely be using a ratio to set up a proportion. The word *combined* 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 Scottsdale residents and of Wharton residents. We are given the value of one quantity, and we are looking for the total of the two quantities. This is a part-to-whole problem. Since we are looking for the total number of residents, we must set up a proportion that compares the ratio of Scottsdale residents to the total number of residents to the ratio of actual Scottsdale residents to the total number of actual residents.

*Write number sentences for each operation*.

The ratio of Scottsdale residents to Wharton residents is 7:12, which means that the ratio of Scottsdale residents to the total number of residents is 7:(12 + 7), which is 7:19, or . Use *x* to represent the number of total residents, since that number is unknown. There are 8,435 Scottsdale residents, so the ratio of Scottsdale residents to total residents is 8,435:*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*:

7*x* = (8,435)(19), 7*x* = 160,265, *x* = 22,895. If 8,435 people live in Scottsdale, then 160,265 people live in Scottsdale and Wharton combined.

*Check your work*.

The ratio of Scottsdale residents to the total number of residents is 7:19, so the ratio of actual Scottsdale residents to the total number of actual residents should reduce to 7:19. The greatest common factor of 8,435 and 22,895 is 1,205, and 8,435:22,895 reduces to 7:19.

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