Solving a word problem using a system of linear equations of the form Ax+By=C

Question

Boris 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 Monday there were 3 clients who did Plan A and 5 who did Plan B. On Tuesday there were 12 clients who did Plan A and 2 who did Plan B. Boris trained his Monday clients for a total of 9 hours and his Tuesday clients for a total of 9 hours. How long does each of the workout plans last?

Answer 1

Let time taken by plan A be x. And time taken by plan B be y.

Given,

3x + 5y = 9 ------------------------(1)

12x + 2y = 9  -----------------------(2)

On (1)*4 - (2) we get:

18y = 27

or y = 27/18 = 3/2 = 1.5

On putting the value of y in eq(1) we get:

3x + 5(3/2) = 9

or 3x = 9 - 15/2

or x = 3/2 * 1/3

or x = 1/2

So, Plan A takes half an hour and Plan B takes 1 hour 30 minutes

Answer 2

Workout Plan A lasts A hours while workout Plan B lasts B hours.

On Monday: 3A+5B=9; on Tuesday: 12A+2B=9.

Multiply Monday’s by 4: 12A+20B=36, so 12A=36-20B.

So we can substitute for 12A on Tuesday:

36-20B+2B=9, 36-18B=9.

Divide through by 9:

4-2B=1, 2B=3, so B=3/2=1.5 hours.

So, since 12A+2B=9, 12A+3=9, 12A=6, A=6/12=0.5 hours.

Plan A is 0.5 hours (30 mins) and Plan B=1.5 hours (90 mins).