A bookstore offers new books at $15 each and used books at $10 each.

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

M22-12

IanStewart · Answer

Answer D should read 39, not 29.

I can suggest a couple of systematic ways to look at this, though I think it's practical to get the answer within two minutes by guessing-and-checking. First you might notice that the number of new books sold must be odd; otherwise the total sales in dollars would end in 0. You could then take an algebraic approach. If n is the number of new books and u the number of used books, we know that 15n+10u = 125, or dividing by 5, we have:

3n + 2u = 25

We want to know the value of 5n + 2u, which is the profit in dollars. Notice how similar this is to the left side of the equation above:

5n + 2u = 2n + (3n + 2u) = 2n + 25

So we just want to find what values are possible for 2n + 25. Remembering that n must be odd, it's easy enough just to plug in n=1, 3, 5 and 7 to see that every answer choice is possible except for 41.

Or, if you know that n is odd, you can replace it with 2k + 1, for some integer k. Then the quantity we're trying to find becomes

2u + 25 = 2(2k+1) + 25 = 4k + 27 = 4k + 24 + 3 = 4(k+6) + 3

So our profit is 3 greater than a multiple of 4, and thus has a remainder of 3 when divided by 4. Thus 41 is impossible (you might, when looking at the answers, see that 41 is a bit suspicious - all of the answer choices give a remainder of 3 when divided by 4 with one exception - 41). That's probably more work than the first approach, though it's perhaps interesting to see why each answer has the same remainder by 4.

Actually, the first approach I took when looking at the question was to treat it something like a weighted average. If the store only sells new books, then it makes one third of a dollar profit for each dollar of sales. If it only sells used books, it makes one fifth of a dollar profit for each dollar of sales. So if it sells a combination of new and used books for S dollars, the profit must be somewhere between S/5 and S/3. We know the total sales was $125, so the profit must be between $125/5 and $125/3, or in other words, between $25 and $41.67. Unfortunately that doesn't rule out any answer choices right away, but the answer $41 is suspiciously close to the maximum here (remember we get the max if the store *only* sells new books, and we know the store sold some used books as well since $125 is not a multiple of 15, so $41 seems very unlikely), so I'd be nearly certain $41 was impossible. I'd then find the profit if the store sold as many new books as possible to verify that their max profit was $39, not $41.

toshio86 · Answer

I guess the fastest way could be to calculate the maximum possible profit, which is 7*5+1*2= 37. Now you know that 41 can't be the answer!

If answer D is actually 39 as IanStewart says, just rule it out...

IanStewart · Answer

The max profit is 7*5 + 2*2 = 39, and not 37; 37 is actually impossible here.

That's certainly very fast, but I suppose the question is - how do you know to do this instead of finding the minimum profit? And what would you do with different answer choices - say all of the answers were between 27 and 39? It's for that reason that I discussed a few ways to look at the question, but I do think a kind of 'plug in numbers and see what happens' approach is perfectly good here, quite practical to do within 2 minutes.

Bunuel · Answer

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

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.

pretzel · Answer

Hi Bunnel, why do we have 2n + (3n+2u)?

Bunuel · Answer

We break 5n+2u into 2n and (3n+2u) because we know the value of 3n+2u (25), so we can substitute, which further will help to answer the question.

KarishmaB · Answer

Cost of new books is $15 and old is $10. Total sale price is $125. This tells me that number of new books sold was definitely odd since the total sale price ends with a 5. Had number of new books sold been even, the total sale price would have been a multiple of 10. So there must have been 1, 3, 5 or 7 new books and rest old books. So profit would have been an odd multiple of 5 + a multiple of 2.

Number of new and old books could be 1 and 11 (total revenue of 125). Profit 5 + 2*11 = 27
Number of new and old books could be 3 and 8. Profit 15 + 2*8 = 31
Number of new and old books could be 5 and 5. Profit 25 + 5*2 = 35
Number of new and old books could be 7 and 2. Profit 35 + 2*2 = 39

A profit of 41 is not possible.

Answer (E)

bumpbot · Answer

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A bookstore offers new books at $15 each and used books at $10 each.

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