Algebra Help: How to Solve Age Word Problems

If you are or have been an Algebra student, you probably hate the dreaded "math word problem." Not only are these kinds of problems the favorite of Algebra teachers and Algebra textbook writers everywhere, they are also the kind of problems that are likely to appear on SAT, PSAT, and GRE style comprehensive tests.

In this article I discuss how to set and solve math word problems that are centered around two people's ages. (This material is covered in Saxon Algebra 2, Lesson 120.) This kind of math problem will talk about the two individual's ages in the past and/or in the future, then ask you to solve for their current ages. As with most Algebra word problems, if you can set up the math problem correctly with your choice of variables and equations you are two thirds there to finding the answer.

The best way to teach solving age word problems is to work through examples.

Example Age Word Problem #1
A man is 6 times as old as his son. In 4 years he will be 4 times as old as his son will be. How old are they now?

Solution:
As with most Algebra word problems, the choice of variables is very important. Here we need two variables:

M = age of the man now
S = age of the son now

If it helps you to solve these problems easier, you may want to even write the above statements at the beginning of your solution, to remind you what the variables mean. When we are working with times in the past and the future it can get confusing.

We have two equations that we can write based on the word problem. The first statement looks at the present, the second statement looks at the future.

"A man is 6 times as old as his son" can be represented with an Algebraic equation as:

M = 6*S

"In 4 years he will be 4 times as old as his son will be" can be represented in Algebra language as:

M + 4 = 4*(S + 4)

The second equation is where the math student is often tricked up. "M + 4" and "S + 4" represent the man and his son's ages 4 years in the future.

Once we have the two equations we can use substitution to solve this system of equations:

6*S + 4 = 4*(S + 4)
6*S + 4 = 4*S + 16
2*S = 12
S = 6
M = 6*6 = 36

The son is 6 years old now; the man is 36 years old now.

Example Age Word Problem #2
Lydia is twice as old as Hannah. 30 years ago Lydia was 7/2 times as old as Hannah. How old are they now?

Solution:
L = age of Lydia now
H = age of Hannah now

The first statement looks at the present, the second statement looks at the past. The first statement is relatively simple to represent as an Algebraic equation: "Lydia is twice as old as Hannah"

L = 2*H

The second equation is where many math students could make a mistake. "30 years ago Lydia was 7/2 times as old as Hannah" would be represented as:

L - 30 = (7/2)*(H - 30)

"L - 30" and "H - 30" represent Lydia's and Hannah's ages 30 years in the past.

We use substitution to solve our system of two equations:

2*H - 30 = (7/2)*H - 105
75 = (3/2)*H
H = 50
L = 2*50 = 100

Hannah is 50 years old now; Lydia is 100 years old.

Example Age Word Problem #3
10 years ago Mike was twice Carl's age. 20 years from now Mike will be 95 years younger than three times Carl's age. How old are they now?

Solution:
M = age of Mike now
C = age of Carl now

The first statement looks at the past, the second statement looks at the future. This is an example of probably the most difficult of the age word problems you are likely to work with in Algebra.

Here's how we would represent the first statement using Algebraic language: "10 years ago Mike was twice Carl's age"

M - 10 = 2*(C - 10)

Remember "M - 10" and "C - 10" represent the men's ages 10 years ago in the past.

Here's how we would represent the first statement using Algebraic language: "20 years from now Mike will be 95 years younger than three times Carl's age"

M + 20 = 3*(C + 20) - 95

Remember "M + 20" and "C + 20" represent the men's ages 20 years in the future. If a person is "younger", then he is that many years less, so you would subtract that amount. If a person is "older", then he is that many years more, so you would add that amount.

As with the other two problems, we will want to do substitution to solve our problem. First we need to simplify our first equation:

M = 2*C - 20 + 10 = 2*C - 10

Now we substitute into the second equation:

2*C - 10 + 20 = 3*C + 60 - 95
45 = C
M = 2*45 - 10 = 80

Carl is 45 years old now, Mark is 80 years old now.

Source
Saxon, John H., Jr. Algebra 2: An Incremental Development. Third Edition