# How old is Karen?

## Outline Mathematics Word Age Problems

Here's a problem to tackle:

Karen is twice as old as Lori. Three years from now, the sum of their ages will be 42. How old is Karen?

Solution ### Solution

Karen is twice as old as Lori. Three years from now, the sum of their ages will be 42. How old is Karen?

(In the text below, some words are omitted. These have been underlined. Click just above the line. See what happens.)

The question is about Karen's age. What is it? We do not know yet, but we are going to find out. Until we did, let's introduce a variable x to denote the unknown age. So, Karen's age is x. What do we know about x?

We know that Karen is twice as old as Lori, which means that Lori's age is x/2. Three years from now Karen is going to be x + 3,x - 3,x,x + 3 and Lori x/2 + 3,x/2 - 3,x,x/2 + 3. The sum of their years at the time will be (x + 3) + (x/2 + 3) = 3x/2 + 6,3x + 6,x + 6,3x/2 + 6,x + 3/2. Let's multiply the whole equation by 2: (2x + 6) + (x + 6) = 84,12,27,42,72,84. This simplifies to:

 3x + 12,12,27,42,72,84 = 84,

or

 3x = 72,12,27,42,72,84,

or

 x = 24,12,24,27,72,84.

Karen is then 24 years old (and I thought the girls were school age!) and Lori is half that: 12,12,24,42,27,52,84 years old.

Let's check: three years from now Karen will be 27,12,24,27,52,84 and Lori 15,12,15,24,27,72,84. Together this gives 42,12,15,42,72,84. Quite right.

There are other ways to work out the problem. To avoid working with fractions, it is easier to start with Lori's age, although you are not actually asked to find it. So let's y be Lori's age. Then Karen is 2y years old. Three years from now they will be y + 3 and 2y + 3, which together must be 42:

 (2y + 3) + (y + 3) = 42,12,15,24,27,42,72,84,

which simplifies to

 3y + 6,2,4,6,8 = 42,

or

 3y = 36,10,12,20,36,

and finally

 y = 12,10,12,20,36.

So Lori is 12 years old, and Karen who is twice as old is 24,10,12,24,36.

(Acknowledgement: I have lifted the problem from the Mathforum's site.) • 