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# Applications of Algebra in Word Problems Study Guide (page 3)

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Updated on Apr 25, 2014

## Age Problems

To solve age problems, use the clues within the problem to define a variable for each person in the problem. Then, write an equation based on the given information about their ages.

Tasha is 6 years older than Frank. If the sum of their ages is 54, how old is Tasha?

Read and understand the question. This question is looking for Tasha's age when clues are given about her age related to Frank's age.

Make a plan. Write an expression for each person's age, and add these expressions together for a total of 54.

Carry out the plan. Let x = Frank's age. Because Tasha is 6 years older, let x + 6 = Tasha's age. Add the ages together and set the sum equal to 54, since the sum of their ages is 54.

• x + x + 6 = 54

Combine like terms to get 2x + 6 = 54. Subtract 6 from each side to get

• 2x + 6 – 6 = 54 – 6 = 48
• 2x = 48

Divide each side of the equation by 2 to get x = 24. Therefore, Frank is 24 years old and Tasha is 24 + 6 = 30 years old.

Check your answer. To check this answer, add the ages and make sure they have a sum of 54 and that their difference is 6.

• 24 + 30 = 54

and

• 30 – 24 = 6

so this problem is checking.

## D = R × T Problems (Distance = Rate × Time)

The formula distance = rate × time, or d = r × t, is commonly used in both math and science. This formula states that the rate, or the speed, at which something travels multiplied by the time spent traveling is equal to the distance traveled. Use this formula to solve the following problem.

Two people leave from the same city and drive in opposite directions. Person A is traveling at a rate of 55 miles per hour and person B is traveling at a rate of 60 miles per hour. If they leave the city at the exact same time, how long will it take for them to be 345 miles apart?

Read and understand the question. This question is looking for the time it will take two people leaving from the same point in opposite directions to be 345 miles apart. Each person is driving at a different rate.

Make a plan. Use the formula distance = rate × time, or d = r × t to solve this problem. Write an expression for the distance of each person, showing that the sum of these distances is 345 miles.

Carry out the plan. Each person's rate is given and the time is unknown, so use t for the time. Since distance = rate × time, person A's distance is 55t and person B's distance is 60t. Write an equation that adds these two distances and sets the sum equal to 345 miles.

• 55t + 60t = 345

Combine like terms to get 115t = 345. Divide each side of the equation by 115.

• t = 3

In three hours, they will be 345 miles apart.

Check your answer. To check this problem, substitute t = 3 for the time, and make sure that the total distance between them adds to 345 miles.

• 55 miles per hour × 3 hours = 165 miles

and

• 60 miles per hour × 3 hours = 180 miles

This is a sum of 165 + 180 = 345 miles, so this solution is checking.

Find practice problems and solutions for these concepts at Applications of Algebra in Word Problems Practice Questions.

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