Evan washes two types of vehicles. It takes him 30 minutes to wash a car and 40 minutes to wash a truck. He charges $12 to wash a car and $15 to wash a truck. In 270 minutes, Evan made $105 washing cars and trucks. How many trucks did Evan wash?
30c+40t = 270
12c+15t = 105
now, from the 2nd equation we see that
12c = 105 - 15t, so
30c = 5/2 (105-15t)
plugging that into the 1st equation, we get
5/2(105-15t) + 40t = 270
5(105-15t) + 80t = 540
105-15t+16t=108
t = 3
t=3
To find out how many trucks Evan washed, we can set up two equations based on the given information.
Let's assume the number of cars Evan washed is 'x' and the number of trucks Evan washed is 'y'.
First, let's determine the total time spent washing cars and trucks:
Time taken to wash x cars = 30x minutes
Time taken to wash y trucks = 40y minutes
According to the problem, the total time spent washing cars and trucks is 270 minutes. So we have the equation:
30x + 40y = 270
Next, let's determine the total amount of money Evan made:
Amount made from washing x cars = $12x
Amount made from washing y trucks = $15y
According to the problem, the total amount of money he made is $105. So we have the equation:
12x + 15y = 105
Now we have a system of equations:
30x + 40y = 270 -- equation (1)
12x + 15y = 105 -- equation (2)
We can now solve this system of equations to find the values of x and y.
To make the equations easier to work with, let's multiply equation (2) by 2:
24x + 30y = 210 -- equation (3)
We can then solve equations (1) and (3) simultaneously using either substitution or elimination method.
Multiplying equation (1) by 3, we get:
90x + 120y = 810 -- equation (4)
Now, subtract equation (3) from equation (4):
(90x + 120y) - (24x + 30y) = 810 - 210
90x + 120y - 24x - 30y = 600
Combining like terms, we have:
66x + 90y = 600 -- equation (5)
Now, we have a new system of equations:
66x + 90y = 600 -- equation (5)
24x + 30y = 210 -- equation (3)
We can continue to solve this system of equations using either substitution or elimination method.
Let's use the elimination method to eliminate the y variable:
Multiply equation (5) by 3 and equation (3) by -2:
198x + 270y = 1800 -- equation (6)
-48x - 60y = -420 -- equation (7)
Adding equation (6) and equation (7):
(198x + 270y) + (-48x - 60y) = 1800 - 420
198x - 48x + 270y - 60y = 1380
Combining like terms, we have:
150x + 210y = 1380 -- equation (8)
Now we have a new equation:
150x + 210y = 1380 -- equation (8)
24x + 30y = 210 -- equation (3)
Next, we'll multiply equation (8) by 2 and equation (3) by 7:
300x + 420y = 2760 -- equation (9)
168x + 210y = 1470 -- equation (10)
Subtract equation (10) from equation (9):
(300x + 420y) - (168x + 210y) = 2760 - 1470
300x + 420y - 168x - 210y = 1290
Combining like terms, we have:
132x + 210y = 1290 -- equation (11)
Now, let's rewrite equation (11):
132x + 210y = 1290 -- equation (11)
24x + 30y = 210 -- equation (3)
Now we have a new system of equations:
132x + 210y = 1290 -- equation (11)
24x + 30y = 210 -- equation (3)
Let's multiply equation (3) by 7:
168x + 210y = 1470 -- equation (12)
Now we have a new equation:
132x + 210y = 1290 -- equation (11)
168x + 210y = 1470 -- equation (12)
Subtract equation (11) from equation (12):
(168x + 210y) - (132x + 210y) = 1470 - 1290
168x - 132x + 210y - 210y = 180
Combining like terms, we have:
36x = 180
Divide both sides of the equation by 36:
x = 5
So, Evan washed 5 cars.
To find the number of trucks Evan washed, substitute the value of x into equation (3):
24(5) + 30y = 210
120 + 30y = 210
Subtract 120 from both sides of the equation:
30y = 90
Divide both sides of the equation by 30:
y = 3
Therefore, Evan washed 3 trucks.