Diesel or Gasoline Engine?

Date: 10/16/96 at 23:24:59
From: Nick Fuller
Subject: Diesel or Gasoline Engine?

A new car may be purchased with either a gasoline or a diesel engine.  
The diesel engine gets 32 mpg.  The gasoline engine gets 29 mpg.  
Gasoline cost $1.16 per gallon and diesel costs $1.04 per gallon.  

The diesel engine costs $427.65 more than the gasoline engine.  How 
far must you drive a car with a diesel engine before the savings in 
fuel pay for the higher price of the engine?

I've have came up with a couple of answers but don't think they are 
correct.  I was wondering if you could help me out with something to 
check my answer with.

Date: 10/17/96 at 9:50:36
From: Doctor Keith
Subject: Re: Diesel or Gasoline Engine?

Hi Nick,

You've got a nice problem.  When facing a word problem it is always
a good idea to try to get a feel for what is going on and what you
think you should get as an answer.  To help us, let's make a table of
what we know:

                          gas car       diesel car
miles per gallon(mpg)       29             32
price of fuel              $1.16          $1.04
additional cost            $0             $427.65

Notice a few things:

  1) the diesel car will go farther on one gallon of fuel.
  2) the diesel fuel costs less.
  3) the diesel car costs more.

So we know the diesel car costs more at first but as you drive it will
cost less per mile than a gas car due to the mpg and the fuel cost.
Thus at some point we expect the  diesel car to be cheaper, counting 
the fuel savings.  Since the fuel and the mpg are not that different 
and the cost of the engine is much more than the difference in the 
costs, we expect it to take a long time to get our money back.  
    
Why bother doing this?  Well, first off it helps organize our 
thoughts. Second, it checks the problem and our comprehension of the 
problem. Third, we have some idea about what we should get (a big 
number). So it is a good idea to take some time to check before 
starting a problem.

Okay, we are ready to start the problem.  First we are looking for the 
miles we must drive until we pay off the $427.65 for the diesel 
engine. 

Let's call the number of miles we will have to drive x.  Now we want 
to write expressions for how much it will cost for the gas and diesel 
cars to drive x miles.  We will do this in two steps so you get the 
feel of it:

Step 1: get the gallons of fuel we will use in x miles


For the gas car:    
                                             (gallons)     x 
(x miles)/(29 miles per gallon)= (x miles) x ---------- = ---  gallons
                                             (29 miles)   29


For the diesel car:    
                                             (gallons)     x 
(x miles)/(32 miles per gallon)= (x miles) x ---------- = ---  gallons
                                             (32 miles)   32



Step 2: get the cost for that much fuel

gas:    ((x/29) gallons)(1.16 dollars per gallon)= x(1.16/29) dollars

diesel: ((x/32) gallons)(1.04 dollars per gallon)= x(1.04/32) dollars

Now that we know how much it will cost in fuel to drive both x miles, 
we need to make one final observation.  When we defined x we said it 
was the miles we had to drive so that we paid off the additional cost 
of a diesel engine.  Thus we know that the cost of driving the gas car 
x miles is equal to the cost of driving the diesel car plus the cost 
of the diesel engine.  In a math equation this is:

x(1.16/29) = x(1.04/32) + 427.65

So all we need to do is solve for x.

             x(0.04) = x(0.0325) + 427.65
 x(0.04) - x(0.0325) = 427.65
x((0.04) - (0.0325)) = 427.65
           x(0.0075) = 427.65
                   x = 427.65/0.0075
                     = 57,020 miles

That is a lot of driving.  Just as a frame of reference the average 
car gets driven 12,000 miles a year.  Thus it will take us about four 
years and nine months of average driving to recover the initial 
investment!  That is quite a while in my book.  Hope this helps.

-Doctor Keith,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/