M22-12

Question

A bookstore offers new books at $15 each and used books at $10 each. For every new book sold, the store earns a profit of $5, whereas for each used book, the profit is $2. If the bookstore's total sales for a particular day amounted to $125, which of the following values cannot be the profit for that day?

A. $27
B. $31
C. $35
D. $39
E. $41

Show Answer

Official Answer

E

A. $27
B. $31
C. $35
D. $39
E. $41

Bunuel · Accepted Answer

Official Solution:

A bookstore offers new books at $15 each and used books at $10 each. For every new book sold, the store earns a profit of $5, whereas for each used book, the profit is $2. If the bookstore's total sales for a particular day amounted to $125, which of the following values cannot be the profit for that day?

A. $27
B. $31
C. $35
D. $39
E. $41

Show Answer

Official Answer

E

A. $27
B. $31
C. $35
D. $39
E. $41

We are given the equation  15n + 10u = 125 , where  n  represents the number of new books sold and  u  represents the number of used books sold.
 
The question is: considering the above equation, which value is not possible for  5n + 2u ?
 
Dividing the equation  15n + 10u = 125  by 5, we get  3n + 2u = 25 . From this, it's evident that  n  must be an odd number. If  n  were even, the equation  3n + 2u  would sum to an even value, which cannot be equal to the odd number 25.
 
Next,  5n+2u=2n+(3n+2u)=2n+25 . Now, you can see that if  n  is 1, 3, 5, or 7, then options A, B, C, and D are possible. Otherwise, note that for E to be possible,  n  must be 8, which is even, and we know that  n  is odd.

Answer: E

rishi02 · Answer

A bookstore sells new books for $15 each and used books for $10 each. On every new book, the store makes a profit of $5 while on every used book it makes a profit of $2. If on a given day the bookstore's sales amounted to $125, which of the following cannot be the profit made on that day?

A. $27
B. $31
C. $35
D. $39
E. $41

Alternate solution :

Try to find the max and min profit

Min profit will happen when the store sells maximum used books i.e 11 used books and 1 new book 
Min profit = 22+5 =27$

Max profit will happen when the store sells maximum new books i.e 7 new books and 2 old books
Max profit = 35 + 4 = 39$

Therefore it is not possible to make a profit of 41$

thiagod · Answer

My solution is very similar to that of Bunuel, but since the reasoning is slightly different, I decided to post it.

Let x be the number of NEW books sold, and y be the number of USED books sold. Then, we'll have:

15x + 10y = 125

Analyzing this equation, you can notice that x MUST be an odd value for the equation to hold true; otherwise, the ending 5 digit of the 125 total sales would disappear. For example, if x=2, then 15x=30, and regardless of the y value, the last digit of total sales value would be 0. Not good.

So, our mission is to find the total profit number that will produce an EVEN value for x. Let the profit equation be: 5x + 2y = A (A=profit).

15x + 10y = 125 (divide the whole equation by -5, so we can eliminate y value, since we're not interested in it)
Sum the new total sales equation with the profit equation:
-3x -2y = -25
5x + 2y = A

This is our final equation: 2x = A - 25.
Quickly test the alternatives (replacing by A) to see which one yields an EVEN x (EmpowerGMAT suggests that in "which of the following" questions, we start by testing either D or E, since those are more likely to be the correct answer). You will find that alternative E yields x=8, while all others yield ODD values for x.

Answer: E

It's not a 2-minutes question, but if you have time in the GMAT day maybe it's worth spending a little extra time on it

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Chrisharley · Answer

I like the solution - it’s helpful.

DonBosco7 · Answer

Hi  Bunuel

Please clarify whether this reasoning is correct for this sentence in the question  For every new book sold, the store earns a profit of $5, whereas for each used book, the profit is $2

- if i say no. of new books sold is  a  and no. of old book sold is  b 
then the portion is bold can it be inferred as  Profit per new book i.e Profit/a = 5 
and,  Profit per old book i.e profit/b = 2

Cause whenever i see the statements like these -> for every X there is Y .............. I think of ratios
so originally i thought of it as new_book/profit = 5 and old_book/profit = 2

Bunuel · Answer

Your reasoning is incorrect.

The phrase "For every new book sold, the store earns a profit of $5" means the profit per new book is $5. Similarly, "For each used book, the profit is $2" means the profit per used book is $2.

You can't infer ratios like new_book/profit = 5 or old_book/profit = 2. The correct interpretation is:

Profit for each new book = $5 
 Profit for each used book = $2

In other words, the profit per book is fixed, not a ratio between the number of books and the profit. For more, please study the OE above.

M22-12

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