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# 700 level question

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700 level question [#permalink]  21 Aug 2010, 22:52
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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.29

E.41
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Re: 700 level question [#permalink]  21 Aug 2010, 23:42
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shekar123 wrote:
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.29

E.41

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.
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Re: 700 level question [#permalink]  22 Aug 2010, 18:17
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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...
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Re: 700 level question [#permalink]  22 Aug 2010, 19:40
toshio86 wrote:
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...

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.
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Re: 700 level question [#permalink]  22 Aug 2010, 20:23
i couldn't find any combinations that can fit into 29 and 41, so i guess D and E are the answers unless there's a typo for any of them.
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Re: 700 level question [#permalink]  23 Aug 2010, 00:05
Profits can only be 25, 27, 39, 48 dollars!
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Re: 700 level question [#permalink]  29 Aug 2010, 20:55
New books revenues can be either 15,45,75,or 105 and corresponding used book revenues would e 110,80,50 and 20 making profits will be 27,31,35,39. hence answer is E. if D is typo
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Re: 700 level question [#permalink]  30 Aug 2010, 00:41
D should be 39 and not 29....E is the clear answer then
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Re: 700 level question [#permalink]  16 Jun 2011, 01:09
3u + 2n = 25
u(max) = 8
check for u =7,5,3,1

will help in POE
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Re: 700 level question   [#permalink] 16 Jun 2011, 01:09
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