Carcass wrote:
A merchant paid $300 for a shipment of x identical calculators. The merchant used 2 of the calculators as demonstrators and sold each of the others for $5 more than the average (arithmetic mean) cost of the x calculators. If the total revenue from the sale of the calculators was $120 more than the cost of the shipment, how many calculators were in the shipment?
A. 24
B. 25
C. 26
D. 28
E. 30
APPROACH #1: Testing the answer choicesSTRATEGY: As with all GRE Problem Solving questions, we should immediately ask ourselves, Can I use the answer choices to my advantage?
In this case, it's possible to test the answer choices, AND I can see how we might eliminate some answer choices first (more on this later).
Now we should give ourselves about 20 seconds to identify a faster approach.
Well, we can also solve the question algebraically, but that approach could take a while for this tricky question.
Since I feel like I can eliminate 2 or 3 answer choices first, I'm going to test the given values. Given: The merchant sells EACH of the remaining calculators for $5 more than the average cost of the x calculatorsSo, for example, if the correct answer is C (x = 26) , then the average cost of each of the 26 calculators will be a complete mess, since 26 doesn't divide nicely into $300
We get: average cost per calculator = $300/26 = $11.538461...
This means the merchant would have to sell each of the remaining 24 calculators at a price of $16.538461..., which makes no sense.
Since answer choices A, C and D don't divide nicely into $300, I'm not going to test these values.
Aside: If the merchant bought 24 calculators (answer choice A), the average price per calculator would be $12.50, which isn't a bad to work with. So if answer choices B and E don't work out, I'll know the correct answer must be A. Let's test answer choice B (
x = 25)
If it costs $300 to purchase 25 calculators, then the average cost per calculator = $300/25 = $12
This means the merchant will sell each of the remaining 23 calculators at a price of $17
So, the total revenue = (23)($17)
I already know this answer is incorrect, because the question tells us that the total revenue was $120 more than the cost of the shipment. In other words, the total revenue must be $420, and since (23)($17) definitely does not equal $420, we can eliminate answer choice B.Now let's test answer choice E (
x = 30)
If it costs $300 to purchase 30 calculators, then the average cost per calculator = $300/30 = $10
This means the merchant will sell each of the remaining 28 calculators at a price of $15
So, the total revenue = (28)($15) = $420
Voila!!
The correct answer is E.
APPROACH #2: AlgebraThe merchant originally buys x calculators for $300
So, the average purchase cost =
300/x dollars per calculator
Later, the calculators are sold for
$5 more than the average purchase cost of
300/x dollars
So, the resell price is
(300/x) + 5 dollars per calculator
How many were sold?
Well, the merchant began with x calculators, but used 2 as demonstrators, which means the merchant sold
x - 2 calculators.
Finally, the merchant's profit was $120 (after the $300 investment). So, the total revenue was
$420We can now write an equation:
[(300/x) + 5](x - 2) = 420IMPORTANT: This is an awful equation to solve. At this point, it may be faster to try plugging in the answer choices.
Or we can solve the equation as follows:
Start with:
[(300/x) + 5](x - 2) = 420 Expand:
300 - (600/x) + 5x - 10 = 420Multiply both sides by x to get:
300x - 600 + 5x² - 10x = 420xSimplify:
5x² - 130x - 600 = 0Divide both sides by 5 to get:
x² - 26x - 120 = 0Factor:
(x - 30)(x + 4) = 0So,
x = 30 or
x = -4Since x can't be negative, we know that
x = 30Answer: E
_________________
Brent Hanneson - founder of Greenlight Test Prep