SOLUTION: Miguel 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 4 clients who

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Question 1201892: Miguel 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 4 clients who did Plan A and 8 who did Plan B. On Tuesday there were 2 clients who did Plan A and 3 who did Plan B. Miguel trained his Monday clients for a total of 9 hours and his Tuesday clients for a total of 4 hours. How long does each of the workout plans last?
Found 3 solutions by ikleyn, Theo, greenestamps:
Answer by ikleyn(52813)   (Show Source): You can put this solution on YOUR website!
.
Miguel 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 4 clients who did Plan A and 8 who did Plan B.
On Tuesday there were 2 clients who did Plan A and 3 who did Plan B.
Miguel trained his Monday clients for a total of 9 hours and his Tuesday clients for a total of 4 hours.
How long does each of the workout plans last?
~~~~~~~~~~~~~~~~~~~ 

Let x = hours for plan A and y = hours for plane B.

Write equations as you read the problem

    4x + 8y = 9     (1)   (total hours on Monday)
    2x + 3y = 4     (2)   (total hours on Thursday)  


To solve this system, multiply equation (2) by 2 (both sides).
Keep equation (1) as is.  You will get

    4x + 8y = 9     (3)   
    4x + 6y = 8     (4)  


Now subtract equation (4) from equation (3), The terms with "4x" will cancel each other,
and you will get

         8y - 6y = 9 - 8

            2y   =   1

             y   = 1/2 = 0.5.


Then from equation (2)

    2x + 3*0.5 = 4,

    2x + 1.5   = 4

    2x         = 4 - 1.5 = 2.5

     x                   = 2.5/2 = 1.25.


ANSWER.  Plan A is 1.25 hours per client (same as 1 hour and 15 minutes).

         Plan B is 0.5  hours per client (same as 30 minutes).

Solved.

------------------

On the way, you learned on how the Elimination method works,
when you solve systems of two equations in two unknowns.


/////////////////


Notice that the way on how tutor @Theo introduces his unknown variables may perplex/confuse you.

Surely, x and y in his post are not the numbers of clients;
they are the hours per a client for plan A and plane B, respectively.

Precisely as they are introduced in my solution.



Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
x = number of clients for plan A.
y = number of clients for plan B.
you have two equtions that need to be solved simultaneouly.
they are:
4x + 8y = 9 on monday.
2x + 3y = 4 on tuesday.
multiply both sides of the first equation by 2 and leave the first equatio as is to get:
4x + 8y = 9
4x + 6y = 8
subract the second equation from the first to get:
2y = 1
solve for y to get:
y = 1/2 = .5
replace y with .5 in the first equation to get:
4x + 8y = 9 becoms 4x + 4 = 9 which becomes 4x = 5 which results in x = 1.25.
you have:
x = 1.25
y = .5
replace a and y with those values in the original equations to get:
4x + 8y = 9 becomes 4 * 1.25 + 8 * .5 = 9 which becomes 5 + 4 = 9 which is true.
2x + 3y = 4 becomes 2.5 + 1.5 = 4 which is also true.
the values of x and y are confirmed to be good.
your solution is:
plan A workout session lasts 1.25 hours.
plan B workout session lasts .5 hours.
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


Knowing how to set up and solve the problem using formal algebra is certainly a good skill to have.

But you can get good mental exercise, and excellent problem-solving experience, by solving the problem informally, using logical reasoning and simple arithmetic.

Suppose Miguel's business on Tuesday doubled, so that he had 4 clients on plan A and 6 on plan B, for a total of 8 hours.

Then compare that to Monday's work, where there were 4 clients on plan A and 8 on plan B, for a total of 9 hours.

Then the difference between the two days would be 2 more clients on plan B, with the same numbers on plan A for both days, with the time difference being 1 hour.

So the time for the two additional clients on plan B is 1 hour, which means each workout on plan B last half an hour.

Then use that with simple arithmetic with the workouts from either day to find that each workout on plan A lasts 1.25 hours.
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