Video Explanation

Text Explanation

Age problems typically involve keeping track of a lot of information. It can be helpful to organize that information into different time periods. If we let the variable S represent Samir's present age and the variable D represent Deepak's present age, we can then think about their past and future ages in the following way:

These expressions for the different time periods will enable us to construct equations based on the information in the prompt.

"7 years ago, Samir was 3 times as old as Deepak" =
(S − 7) = 3(D − 7)
S − 7 = 3D − 21
S = 3D − 14

"In 4 years, Samir will be twice as old as Deepak" =
(S + 4) = 2(D + 4)
S + 4 = 2D + 8
S = 2D + 4

We now have a system of equations that we can solve via either the substitution or elimination methods. Using substitution, we get:

3D − 14 = 2D + 4
D − 14 = 4
D = 18

Deepak's present age is 18.

FAQ: Can I use a single variable approach?

Yes, you can, but it's not very intuitive. Age problems are tricky enough already when different times are involved, and if we tried to streamline these time periods into one variable, it would get more complicated still. That might be the best way to identify when you'd want to/be able to use the one-variable approach. If you can't immediately see how to use a single variable, then don't.

If you did use a one-variable approach, you'd immediately come to some trouble in trying to define Samir's age using D because the only info we have is from other time periods. The best way to tackle this, then, would be to make D = Deepak's age 7 years ago, which has a clearer relationship to Samir's age back then.  Then we'll just add 7 at the end to get Deepak's present age.

We can start by again making a table that represents the two ages at each point in time:

7 years ago
Deepak's age: D
Samir's age: 3D

Present
Deepak: D + 7
Samir: 3D + 7

4 years from now
D's age: D+11
S's age: 2(D+11)

Now we just need to represent this with an equation. What is one more piece of information that we know that isn't represented by this table? Well, we know that people age at the same rate. So Samir's age 4 years from now will be 11 more than his age was 7 years ago. So we can write:

3D + 11 = 2(D + 11)
And then we can solve for D:
3D + 11 = 2(D + 11)
3D + 11 = 2D + 22
D + 11 = 22
D = 11

Then, remember that D = Deepak's age 7 years ago. So that means 11 + 7 = 18, his age right now.