Finding a difference between averages

[Victor Amelkin]

Hello,

 

I recently encountered a simple math problem with the solution of which I do not quite agree. Could you please suggest whether my doubts concerning this solution are grounded?

 

-- Problem -------------------------------------------------------------------------

 

Data:

 

* Standard car models sold: 30,000

* Basic car models sold: 15,000

* Total revenue from sales of the standard model exceeded total revenue from sales of the basic model by $41 million.

 

Question:

 

By approximately how much dollars does the average sales price of a standard model exceeded the average price of a basic model?

 

-- "Solution" -----------------------------------------------------------------------

 

Actually, it is a problem from one GRE guide, which provides the following solution to it:

 

delta_n = "difference in numbers of sold models" = 15,000

delta_$ = difference in total revenues" = 41,000,000$

 

=>

 

delta_avg = "difference in average prices of standard and basic models" = delta_$ / delta_n ~ 2,700$

 

-- "Solution" does not work ---------------------------------------------------

 

However, I do not quite understand how this calculation may solve such a problem. Here is an example for which this solution does not work:

 

Model #1:

 

n1 (number of cars sold) = 10

avg1 (average price per model) = 5$

total1 (total revenue per model) = n1 * avg1 = 10 * 5$ = 50$

 

Model #2:

 

n2 (number of cars sold) = 5

avg2 (average price per model) = 2$

total2 (total revenue per model) = n2 * avg2 = 5 * 2$ = 10$

 

Solution:

 

delta_n = n1 - n2 = 5;

delta_$ = total1 - total2 = 40$

 

According to the solution from the GRE guide:

 

delta_avg = delta_$ / delta_n = 40$/5 = 8$

 

However, delta_avg = avg1 - avg2 = 5$ - 2$ = 3$ != 8$.

 

-- Finale ----------------------------------------------------------------------------

 

The solution from the guide seems to be wrong, but maybe it is me who is missing something? =)

 

Thanks in advance.

 

--

Victor

 

P.S. As for me, the answer to the original problem as it is stated should be "It cannot be determined from the data given".

[sueka]

I believe it should be D. i find your examples to completely relevant.

[kevinp123]

Look at it this way. 30,000*a(1) - 15,000*a(2) = 41,000,000, where a(1) and a(2) are the average prices of each model. Divide the equation by 30,000 to get a(1) - (1/2)*a(2) = 1366, however this is going to overstate the actual difference in average prices because you are only subtracting 1/2 of the lower average price. In the counter example you provided, the equation will be 10*a(1) - 5*a(2) = 40 which comes to a(1) - (1/2)*a(2) = 4, which overstates $3 by a little bit. However, to get an exact answer, you would need to know either a(1) or a(2).

 

I think this is a pretty bad question, but its one saving grace might be the fact that it say "approximately", even though the solution it poses is not even "approximately" close.

[paul2432]

If the question is a quantitative comparison, then it may be possible to give an answer of A or B depending on the answer choices. For example if the choices were:

 

A: The difference in average price

B: 1300

 

The answer would be A. Using Kevin's nomenclature we would get a(1) - a(2) = 1366 + 1/2 x a(2). The difference must be larger than 1300. Similarly we could divide Kevin's equation by 15,000 instead of 30,000 to get a(1) - a(2) = 2733 - a(1)

 

Then if the answer choices were:

 

A: The difference in average price

B: 2800

 

The answer would be B.

 

I agree that the solution presented by the book is lacking.

 

Paul