Question 91914: Car rental agency A will rent a compact car for $35 per day and an additional charge of $0.24 per mile. Car rental agency B will charge only $0.16 per mile but charges $41 per day. If Adam wanted to rent a car for three days, how many miles would Adam have to drive to make car rental agency B a better bargain?
By trial and error I have found that the answer is 226 but can't figure out an equation, please help.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! To keep this in notation that you are probably most familiar with, let's represent the cost
of renting the car with the letter y. And let's represent the number of miles driven with
the letter x.
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Adam is going to rent a car for three days. Since agency A charges $35 per day in fixed costs,
Adam will owe that company $35 times 3 or $105 dollars ... not including the mileage
charges. The mileage charges are $0.24 for every mile driven, and since x represents the
number of miles he drives, he will owe agency A $0.24 times x or $0.24x in mileage charges.
The total amount y that he will owe to agency A for the three day rental isis the sum of
the fixed costs of $105 plus the mileage charge of $0.24x. So we can write the equation for
the total dollars y that he will pay to agency A as:
.
y = 0.24x + 105
.
Now let's do a similar analysis for agency B. At $41 per day, the 3 day rental will cost
Adam $41 times 3 which is $123 in fixed costs. And at $0.16 per mile for the x miles that
Adam drives he will owe agency B $0.16 times x or 0.16x in mileage charges. The total cost
y charged by agency B is the sum of these two or:
.
y = 0.16x + 123
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So you now have two equations to work with as follows:
.
y = 0.24x + 105
y = 0.16x + 123
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If you solve these equations simultaneously you find the solution that is the "break even"
point ... that is the solution where the total cost to both agencies is the same.
.
Let's do that ...
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There are a couple of ways this could be done ... variable elimination or substitution
among them. Let's use substitution. In the first equation we have that y is equal to
0.24x + 105. Suppose we substitute that value for y in the second equation. In that case
the second equation becomes:
.
0.24x + 105 = 0.16x + 123
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Get rid of the 0.16x on the right side by subtracting 0.16x from both sides:
.
0.08x + 105 = 123
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Then get rid of the 105 on the left side by subtracting 105 from both sides:
.
0.08x = 18
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Solve for x by dividing both sides by 0.08 to get:
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x = 18/0.08 = 225
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So if Adam drives 225 miles in the three days he can use either agency and the cost should
be the same.
.
Let's check that out. At agency A he will pay $35 dollars a day for 3 days plus 225 miles
times 0.24 per mile, a total of $105 plus $54 = $159
.
At agency b he will pay $41 per day for 3 days plus 225 miles times 0.16 cents per mile,
a total of $123 + $36 = $159.
.
Just as we suspected. The costs at 225 miles are identical for either of the two agencies.
.
Since agency B charges just $0.16 per mile, Adam will save money by going with agency B
if he drives more than 225 miles.
.
Hope this helps you to understand a mathematical way of coming up with the answer that
you managed to work out by just using your thought power. Good exercise for you!
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