Greg the trainer has two solo workout plans that he offers his clients: Plan A and Plan B. Each client does either one or the other (not both). On Wednesday there were 3 clients who did Plan A and 5 who did Plan B. On Thursday there were 9 clients who did Plan A and 7 who did Plan B. Greg trained his Wednesday clients for a total of 6 hours and his Thursday clients for a total of 12 hours. How long does each of the workout plans last?

This problem can be solved by algebraic method.

Let

x = the total time spent of all clients in Plan A

y = the total time spent of all clients in Plan B

We represent two variables x and y because there are two plans that won't be happened simultaneously.

On Wednesday, the two workout plans have the total time of 6 hours. We equate

3x + 5y = 6

While on Thursday, the total time is 12 hours. We also equate

9x + 7y = 12

To find x and y, we can use the substitution method. For the first equation, we arrange it in terms of y, that is

5y = 6 - 3x

y = (6 - 3x) / 5

Substitute it to the second equation:

9x + (7/5) (6 - 3x) = 12

9x + (42/5) - (21/5) x = 12

Multiply the equation by 5 to cancel the denominator:

45x + 42 - 21x = 60

45x - 21x = 60 - 42

24x = 18

x = 18/24 = 3/4 hours

For y:

3 (3/4) + 5y = 6

9/4 + 5y = 6

Multiply the equation by 4 to cancel the denominator:

9 + 20y = 24

20y = 24 - 9

20y = 15

y = 15/20 = 3/4 hours

Hence, each workout plans are done within 3/4 hours (or 45 minutes).