Question

2. Suppose you are considering buying a Toyota Prius, which starts at $25,000 and gets 50

mpg. A similarly equipped Honda Accord will start at $22,000 but will get 31 mpg.
(a) How many miles will it take the Prius to recoup the price difference?

Answer 1

If a gallon of gas costs $2.9

The Prius can drive 50 miles with one gallon.

then for m miles, it needs:

(m/50) gallons.

and each gallon costs $2.9

Then for driving m miles, the cost would be:

(m/50)*$2.9

Then the total cost of buying the Prius and driving for m miles is:

P(m) = $25,000 + (m/50)*$2.9

We can do a similar approach for the Accord's case.

The Accord can drive 31 miles with one gallon.

Then if it drives m miles, it will need:

(m/31) gallons.

And knowing that each gallon costs $2.9, the cost for driving m miles is:

(m/31)*$2.9

Then the total cost of buying an Accord and driving for m miles is:

A(m) = $22,000 + (m/31)*$2.9

We want to find the value of m, such that the cost of the Prius is smaller than the cost of the Accord.

Then the first thing we need to find is the value of m such that the costs are equal:

P(m) = A(m)

$25,000 + (m/50)*$2.9 = $22,000 + (m/31)*$2.9

$25,000 - $22,000 = (m/31)*$2.9 - (m/50)*$2.9

$3,000 = m*$2.9*(1/31 - 1/50)

$3,000/($2.9*(1/31 - 1/50)) = m = 84,392

This means that if you drive more than 84,392 miles, then the cost of the Prius will be smaller.

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