SOLUTION: The school football team held a car wash to raise money. They washed cars for $4 each and vans for $6 each. They washed a total of 61 vehicles and raised $278. How many of each

Question 1203110: The school football team held a car wash to raise money. They washed cars for $4 each and vans for $6 each. They washed a total of 61 vehicles and raised $278. How many of each type of vehicle did they wash?
Found 3 solutions by math_tutor2020, ikleyn, greenestamps:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

c = number of cars
v = number of vans

c+v = total number of vehicles
c+v = 61

4c = amount of money from the cars only ($4 each)
6v = amount of money from the vans only ($6 each)
4c+6v = total amount of money raised
4c+6v = 278

The system of equations is
c+v = 61
4c+6v = 278

I'll use substitution to solve.
Isolate c in the 1st equation
c+v = 61
c = 61-v

Plug that into the other equation to isolate v.
4c+6v = 278
4(61-v)+6v = 278
244-4v+6v = 278
244+2v = 278
2v = 278-244
2v = 34
v = 34/2
v = 17

Use this to determine c.
c = 61-v
c = 61-17
c = 44

Answers:
Number of cars = 44
Number of vans = 17

Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.
The school football team held a car wash to raise money.
They washed cars for $4 each and vans for $6 each.
They washed a total of 61 vehicles and raised $278.
How many of each type of vehicle did they wash?
~~~~~~~~~~~~~~~~

x washed vans and (61-x) washed cars. Total money equation is 6x + 4*(61-x) = 278. Simplify and find x 6x + 244 - 4x = 278, 6x - 4x = 278 - 244 2x = 34 x = 34/2 = 17. ANSWER. They washed 17 vans and 61 - 17 = 44 cars. CHECK. 6*17 + 4*44 = 278 dollars, total money earned. ! correct !

Solved.

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Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

Look at the responses from the other two tutors who provided formal algebraic solutions to see that it is almost always the case that a tiny bit of extra work at the beginning to set the problem up using only one variable results in a faster and easier path to the solution.

And if formal algebra is not required, this is a very common type of problem that can be solved quickly and easily using logical reasoning and simple arithmetic:

If all 61 vehicles had been cars, the total raised would have been 61($4) = $244.
The actual total was $278, which is $34 more.
The difference between the cost for a car and the cost for a van was $2.
Therefore, the number of vans was $34/$2 = 17.

ANSWER: 17 vans and 61-17 = 44 cars

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