To review these concepts, go to Applications of Algebra in Word Problems Study Guide.

**Applications of Algebra in Word Problems Practice Questions**

**Practice
**

**Problems**

- The sum of three consecutive odd integers is 135. What are the integers?
- Three times a smaller of two consecutive integers is equal to twice the larger integer increased by 27. What are the integers?
- Carolyn is buying two different amounts of candy to form a 6-pound mixture of candy that costs $3 per pound. If one type of candy costs $2.50 per pound, and the other type costs $4 per pound, how much of each type does she need to buy?
- In her change purse, Josie has nickels and dimes. If she has a total of $2.40 and has twice as many nickels as dimes, how many of each coin does she have?
- James is 10 years less than twice as old as Brandon. If the sum of their ages is 56, what are their ages?
- Kaitlin's mother drives her to her grandparent's house 160 miles away and averages 64 miles per hour. On the return trip home, they average only 40 miles per hour because of traffic. How long did the return trip take?

**Solutions**

*Read and understand the question*. This question is looking for three consecutive odd integers that add to 135.*Read and understand the question*. This question is looking for two consecutive integers where clues are given for each.*Read and understand the question*. Carolyn is mixing together two different types of candy. Each type is a different price. You are looking for the number of pounds of each as a final answer.*Read and understand the question*. This question is looking for the total number of nickels and the total number of dimes in a coin purse containing only nickels and dimes. There are twice as many nickels as dimes. Make a plan.Write an expression for the number of dimes and the number of nickels in the bank. Multiply each of these expressions by the value of each type of coin. Then, add these expressions and set them equal to the total of $2.40.*Read and understand the question*. This question is looking for both ages when clues are given about how their ages are related. The sum total of their ages is also given.*Read and understand the question*. This question is looking for the time it took on the return trip. Be aware of extra information. The fact that they averaged 64 miles per hour on the way to Kaitlin's grandmother's house is not necessary to find the solution to the problem.

*Make a plan*. Use the expressions *x*, *x* + 2, and *x* + 4 to define the variables. Then, write an equation by adding these expressions and setting them equal to 135.

*Carry out the plan*. First, let *x* = the smallest odd integer, let *x* + 2 = the next odd integer, and let *x* + 4 = the greatest odd integer. Then, add these expressions together and set them equal to 135. The equation becomes *x* + *x* + 2 + *x* + 4 = 135. Combine like terms to get 3*x* + 6 = 135. Subtract 6 from each side of the equation.

- 3

*x*+ 6 –6 = 135 –6

Simplify to get 3*x* = 129. Divide each side by 3 to get the variable *x* alone.

*x*= 43

Because *x* = 43, then *x* + 2 = 45, and *x* + 4 = 47. The three integers are 43,45, and 47.

*Check your answer*. Check your answer by finding the sum of the three odd integers. The sum is 43 + 45 + 47 = 135, so this problem is checking.

*Make a plan*. Use the expressions *x* and *x* + 1 to define the variables. Then, write an equation by multiplying the smaller integer by 3 and setting this equal to two times the larger plus 27.

*Carry out the plan*. First, let x = the smaller integer and let *x* + 1 = the larger integer. Then, multiply the smaller by 3 to get 3x and increase twice the larger by 27 to get 2(*x* + 1) + 27. Set these values equal to each other. The equation becomes 3*x* = 2(*x* + 1) + 27. Apply the distributive property to get 3*x* = 2*x* + 2 + 27. Simplify to get 3*x* = 2*x* + 29. Subtract 2x from each side of the equation:

- 3

*x*–2

*x*= 2

*x*–2

*x*+ 29

Simplify to get *x* = 29. Because *x* = 29, then *x* + 1 = 30. The two integers are 29 and 30.

*Check your answer*. Check your answer by making sure the two integers fit the clues in the problem. Three times the smaller is 3 × 29 = 87, and twice the larger increased by 27 is 2(30) + 27 = 60 + 27 = 87. Because these values are equal, this problem is checking.

*Make a plan*.Write an expression for the amounts of each type of candy using the information given, and multiply each by the price per pound. Set this equal to the total mixture, which is equal to $3.00 × 6 or 3(6).

*Carry out the plan*. Let *x* = the number of pounds of the $2.50 per pound candy. Since there is a total of 6 pounds in all, then 6 – *x* = the number of pounds of the $4 per pound candy. Next, write an equation that adds the cost of the two types of candy and sets it equal to the total. The cost of the $2.50 candy is 2.5*x*, the cost of the $4 candy is 4(6 –*x*), and the total cost is 3(6). The equation is 2.5*x* + 4(6 – *x*) = 3(6). Simplify by using the distributive property:

- 2.5

*x*+ 24 – 4

*x*= 18

Combine like terms.

- –1.5

*x*+ 24 = 18

Subtract 24 from each side of the equation.

- –1.5

*x*+ 24 – 24 = 18 – 24

Simplify to get –1.5*x* = –6. Divide each side by –1.5:

*x*= 4

Carolyn should buy 4 pounds of the $2.50 candy and 6 – 4 = 2 pounds of the $4 per pound candy.

*Check your answer*. To check this solution, multiply the number of pounds of each by the cost of each and see if it is a total of $3 × 6 = $18. $2.5 × 4 pounds is equal to $10 and $4 × 2 pounds is equal to $8. These amounts have a sum of $10 + $8 = $18, which was the cost of the total mixture. This answer is checking.

*Carry out the plan*. Let *x* = the number of dimes, and let 2*x* = the number of nickels. A dime has a value of $0.10, so multiply 0.10 by *x* to get 0.10*x*. A nickel has a value of $0.05 so multiply 0.05 by 2*x* to get 0.05(2*x*). Next, write an equation by adding these expressions and setting them equal to $2.40. The equation is 0.10*x* + 0.05(2*x*) = 2.40. Multiply to eliminate the parentheses. The equation becomes 0.10*x* + 0.10*x* = 2.40. Combine like terms to get 0.20*x* = 2.40. Divide each side by 0.20:

*x*= 12

Therefore, there are 12 dimes and 2(12) = 24 nickels.

*Check your answer*. Check this solution by making sure that the coins equal a total of $2.40. Twenty-four nickels has a value of $0.05 × 24 = $1.20, and 12 dimes has a value of $0.10 × 12 = $1.20. The sum is $1.20 + $1.20 = $2.40, so this answer is checking.

*Make a plan*.Write an expression for each person's age, and add these expressions together for a total of 56.

*Carry out the plan*. Let *x* = Brandon's age. Because James's age is 10 years less than twice Brandon's age, let 2*x* – 10 = James's age. Add the ages together and set the sum equal to 56, since the sum of their ages is 56:

*x*+ 2

*x*– 10 = 56

Combine like terms to get 3*x* – 10 = 56. Add 10 to each side to get

- 3

*x*– 10 + 10 = 56 + 10

- 3

*x*= 66

Divide each side of the equation by 3 to get *x* = 22. Therefore, Brandon is 22 years old and James is 2(22) – 10 = 44 – 10 = 34 years old.

*Check your answer*. To check this answer, add the ages and make sure they have a sum of 56: 22 + 34 = 56, so this answer is checking.

*Make a plan*. Use the formula *distance* = *rate* × *time*, or *d* = *r* × *t* to solve this problem. Write an expression using the distance of the return trip and the rate to find the time.

*Carry out the plan. *The time is unknown, so use t for the time. Since distance = rate × time, the return trip of 160 miles at a rate of 40 miles per hour gives the equation 40*t* = 160. Divide each side of the equation by 40.

*= 4*

*t*It took 4 hours for the return trip.

* Check your answer. *To check this problem, substitute

*t*= 4 for the time and make sure that the total distance is 160 miles:

- 40 miles per hour × 4 hours = 160 miles

so this solution is checking.

* *

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