Define the variables and set up a system of equations and solve

Question

Katy’s favorite rides at the amusement park are the roller coaster and water slide. The wait time for the roller coaster is 25 minutes and the wait time for the water slide is 10 minutes. Is she went on 12 rides total and waited 3 hours in line, how many times did she go on the ride.

Answer 1

The unknowns are number of times Katy rode the roller coaster and the number of times she rode the water slide. Set number of roller coaster rides to n_R and number of water slide rides to n_W. Total number of rides then is n_R + n_w which is 12.

This gives you one equation n_R + n_w = 12. ----------- equation 1

The total time spent on roller coaster rides is 25n_R minutes, and the total time spent on water slide rides is 10n_W minutes. The total time spent on rides is 25n_R + 10n_w which is 3 hrs (180 minutes).

This gives your second equation 25n_R + 10n_w = 180. -------------- equation 2

You can solve the system of two equations in two variables simultaneously by elimination or by substitution.

Substitution:

From equation one we get n_R = 12 - n_W. Substituting in equation two, we get 25(12 - n_W) + 10n_w = 180. Solve to get n_w = 8. Using equation 1 again n_R = 12 - 8 ----> n_R = 4. Katy rode the roller coaster four times and the water slide eight times.

Elimination:

Arrange the equations one under the other as follows.

n_R + n_w = 12 ----------------------- equation 1

25n_R + 10n_w = 180 ------------------- equation 2

Multiply equation 1 by -10 and add both equations:

-10n_R - 10n_w = -120 ----------------------- equation 1 x -10

25n_R + 10n_w = 180 ------------------------ equation 2

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15n_R = 60 -----------------> n_R = 4.

Solving equation 1 for n_R = 4 we get n_W = 8 like before.

Answer 2

Assume that Katy went for roller coaster x times and water ride for y times.

The total number of rides taken by Katy = 12

=> x+y=12...........(1)

Now she spent 3 hrs or 3*60 = 180 mins waiting for the rides.

=> 25x+10y=180

=> 5x+2y=36..................(2)

Multiplying equation (1) by 2 and then subtracting from eqn. (2), we get

(5x-2x)+(2y-2y)=36-(2*12)

=> 3x=12=> x=4

Putting value of x in equation (1), we get

y=12-4=8

So Katy went on roller coaster for 4 times and water ride for 8 times.