Gary is 8 years older than his brother. Six years ago Gary was three times as old as his brother. How old are they now?

Question

Gary is 8 years older than his brother. Six years ago Gary was three times as old as his brother. How old are they now? 

Answer 1

Let us first introduce some "vocabulary" to translate the information in the problem statement in order to write it down in a mathematical way.

We will use the letter 'g' for Gary's age and 'b' for his brother's age. Both ages are the current ages expressed in years. Then the information provided can be written as:

g = b + 8

g - 6 = 3 * (b - 6)

The problem is thus a system of two linear equations in two unknowns (g & b). Since the first equation is already simplified, we can "plug it into" the second equation which yields:

(b + 8) - 6 = 3 * (b -6)

Simplifying on both sides yields:

b + 2 = 3b - 18

Next, we collect all b's on the right and move everything else to the left:

b + 2 = 3b - 18 | -b

2 = 2b - 18 | + 18

20 = 2b | ÷ 2

10 = b

Thus, b = 10, which put into the original first equation gives us:

g = b + 8 = 10 + 8 = 18.

Gary is 18 years old while his brother is 10.