Age Word Problems Lesson

Over the course of the last few lessons, we have learned how to solve a linear equation in one variable. In real life, our problems will not be given to us in such a manner. We will need to create an equation based on the context of the given situation. When we first learn how to set up and solve word problems, it can be a real challenge for many students. It is often helpful to follow a step-by-step method for solving word problems.

Six-step method for Applications of Linear Equations in One Variable

1. Read the problem carefully and determine what you are asked to find
   • Write down the main objective of the problem
2. Assign a variable to represent the unknown
   • If more than one unknown exists, we express the other unknowns in terms of this variable
3. Write out an equation that describes the given situation
4. Solve the equation
5. State the answer using a nice clear sentence
6. Check the result by reading back through the problem
   • We need to make sure the answer is reasonable. In other words, if asked how many miles were driven to the store, the answer shouldn't be (-3) as we can't drive a negative amount of miles.

Age Word Problems

Age word problems are very common in our study of algebra. This type of word problem generally gives the sum of the ages of family members or friends and asks us to determine the individual ages. Let's take a look at some examples.
Example 1: Solve each word problem.
Two siblings, Jamie and Steven have a combined age of 15. Steven is twice the age of Jamie. Find the age of each sibling.

1. Read the problem carefully and determine what you are asked to find
   • We are asked to find the age of each sibling.
2. Assign a variable to represent the unknown
   • Let x = Jamie's age
   • Then 2x = Steven's age (since it's twice the age of Jamie)
3. Write out an equation that describes the given situation
   • If we add Jamie's age (x) with Steven's age (2x), the result is 15.
   • x + 2x = 15
4. Solve the equation
   • x + 2x = 15
   • 3x = 15
   • 3/3 x = 15/3
   • x = 5
5. State the answer using a nice clear sentence
   • Since x represents Jamie's age, we know that Jamie is 5 years old.
   • Since 2x represents Steven's age, we know that Steven is 2 • 5 or 10 years old
   • Jamie is 5 years old and Steven is 10 years old.
6. Check the result by reading back through the problem
   • We know that the sum of their ages is 15.
   • Check: 5 + 10 = 15
   • 15 = 15
   • We know that Steven is twice the age of Jamie
   • Check: 2 • 5 = 10
   • 10 = 10
   • Since both of our checks were true, we can say our answer is correct.

Example 2: Solve each word problem.
Three coworkers, Beth, Mary, and Alice have a combined age of 87. Mary is two years younger than Beth. In five years, Alice will be twice as old as Mary. Find the age of each woman.

1. Read the problem carefully and determine what you are asked to find
   • We are asked to find the age of each woman.
2. Assign a variable to represent the unknown
   • Let x = Mary's age
   • Then x + 2 = Beth's age (since Beth is two years older)
   • Then 2(x + 5) - 5 = Alice's age (since Alice will be twice as old as Mary in five years)
     • In five years, Mary will be x + 5 years old
     • In five years, Alice is twice as old as Mary 2(x + 5)
     • Subtract five away to get back to today 2(x + 5) - 5
3. Write out an equation that describes the given situation
   • If we add Mary's age (x), with Beth's age (x + 2), and Alice's age (2(x + 5) - 5), the result is 87.
   • x + x + 2 + 2(x + 5) - 5 = 87
4. Solve the equation
   • x + x + 2 + 2(x + 5) - 5 = 87
   • 2x + 2 + 2x + 10 - 5 = 87
   • 4x + 7 = 87
   • 4x + 7 - 7 = 87 - 7
   • 4x = 80
   • 4/4 x = 80/4
   • x = 20
5. State the answer using a nice clear sentence
   • Since x represents Mary's age, we know that Mary is 20 years old.
   • Since x + 2 represents Beth's age, we know that Beth is 22 years old.
   • Since 2(x + 5) - 5 represents Alice's age, we know that Alice is 45 years old.
   • Mary is 20 years old, Beth is 22 years old, and Alice is 45 years old.
6. Check the result by reading back through the problem
   • We know that the sum of their ages is 87.
   • Check: 20 + 22 + 45 = 87
   • 87 = 87
   • We know that Mary is two years younger than Beth
   • Check: Mary is 20 years old, while Beth is 22 years old
   • We know that in five years, Alice will be twice as old as Mary
   • Check: In five years, Alice will be 50 and Mary will be 25
   • Since all of our checks were true, we can say our answer is correct.

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