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Need expert help - Fastest method to solve this question

by voodoo_child » Mon Jul 30, 2012 4:31 am
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?

$27
$31
$35
$39
$41

[spoiler]OA E[spoiler]

Bunuel,
How can I solve this method in two minutes? Any tips? THE OE says "The maximum number of new books that could have been sold is 7. In this case the profit would have been highest: USD. 41 USD is an impossible profit."

If 125 = 15*1 + 11*10=> Maximum number of books = 12 and not 7.

I wrote two equations : 15X + 10y =125; 5X+2Y = (substitute answer choice). However, this took me over 4 minutes.

Thanks
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by niketdoshi123 » Mon Jul 30, 2012 5:44 am
voodoo_child 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?

$27
$31
$35
$39
$41

Bunuel,
How can I solve this method in two minutes? Any tips? THE OE says "The maximum number of new books that could have been sold is 7. In this case the profit would have been highest: USD. 41 USD is an impossible profit."

If 125 = 15*1 + 11*10=> Maximum number of books = 12 and not 7.

I wrote two equations : 15X + 10y =125; 5X+2Y = (substitute answer choice). However, this took me over 4 minutes.
As the answer choices contain different possible profits that could be earned, we can start by calculating the maximum profit which could be earned.

5x + 2y = answer choices ----(1)(Profit earned)
15x + 10y = 125 -----(2) (Total sales amount)

This equation can be further simplified

=> 3x + 2y = 25 ----(2)

To maximize the profit, maximize value of the x
let x = 8
=> 3*8 + 2y = 25
=> y = 1/2 (not possible, as the bookseller can not sell 1/2 book)

let x = 7
=> 3*7 + 2*y = 25
=> y = 2

Plug x = 7 & y = 2 in equation 1 to get the maximum profit

5*7 + 2*2 = 39.

Since the maximum profit made on that day was $39, therefore making a profit of $41 is impossible.
You don't have to solve further.
Option E is the correct answer.
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by Brent@GMATPrepNow » Mon Jul 30, 2012 7:27 am
voodoo_child 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?
$27
$31
$35
$39
$41
OA E
Let N = number of new books sold
Let U = number of used books sold

Sales:
If new books cost $15, and used books cost $10, then the total sales = 15N + 10U
We're told that the total sales is $125. So, we can write: 15N + 10U = 125
We can divide both sides by 5, to get a simpler equation: 3N + 2U = 25

Profits:
If the profit is $5 on new books, and $2 on used books, then the total profits = 5N + 2U

Let's take our two equations...
5N + 2U = profit
3N + 2U = 25
... and subtract the bottom from the top to get:
2N = profit - 25
Or...profit = 2N + 25

Interesting, on this given day, we can see that the profit equals 2N + 25.
Since N must be an integer, we can see that 2N must be even, in which case 2N+25 must be odd.
So, the profit must be an odd number.
When I check the answer choices, I can see that all of the answer choices are odd. Too bad :-(
There must be something else this question is testing.

Okay, let's check the profits.
Answer choice A (the smallest answer choice) suggests that we could get a profit of $27.
Well, if 2N + 25 = profit, then we can write 2N + 25 = 27
When we solve for N, we get N=1.
We can plug N=1 into our equation 3N + 2U = 25 to get U=11.
Yep, that checks out.

Okay, now let's check answer choice E (the biggest answer choice). It suggests that we could get a profit of $41.
Since 2N + 25 = profit, we can write 2N + 25 = 41
When we solve for N, we get N=8.
We can plug N=8 into our equation 3N + 2U = 25 to get U=0.5
This is impossible, so the answer is E

Cheers,
Brent
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by GMATGuruNY » Mon Jul 30, 2012 8:38 am
voodoo_child 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?

$27
$31
$35
$39
$41
15n + 10u = 125.

New books:
For the total amount of revenue to be ODD, 15n -- the number of new books -- must also be ODD.
Each new book generates a profit of $5.
Thus, the following options are possible:
A: 15*1 = 15, profit = 1*5 = 5
B: 15*3 = 45, profit = 3*5 = 15
C: 15*5 = 75, profit = 5*5 = 25
D: 15*7 = 105, profit = 7*5 = 35

Used books:
The BALANCE of the revenue comes for the $10 used books.
Each used book generates a profit of $2.
Thus, for the four options above:
A: (125-15)/10 = 11, profit = 11*2 = 22
B: (125-45)/10 = 8, profit = 8*2 = 16
C: (125-75)/10 = 5, profit = 5*2 = 10
D: (125-105)/10 = 2, profit = 2*2 = 4

Total profit for each option:
A = 5+22 = 27.
B = 15+16 = 31.
C = 25+10 = 35.
D = 35+4 = 39.

Since answer choices A, B, C and D are all possible, the correct answer is E.
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by eagleeye » Mon Jul 30, 2012 5:49 pm
voodoo_child 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?

$27
$31
$35
$39
$41

[spoiler]OA E[spoiler]

Bunuel,
How can I solve this method in two minutes? Any tips? THE OE says "The maximum number of new books that could have been sold is 7. In this case the profit would have been highest: USD. 41 USD is an impossible profit."

If 125 = 15*1 + 11*10=> Maximum number of books = 12 and not 7.

I wrote two equations : 15X + 10y =125; 5X+2Y = (substitute answer choice). However, this took me over 4 minutes.

Thanks
First of all, it is a question that will take time to solve, there is no getting around it. However, I believe using some algebra and some reasoning will lead us to the right answer the fastest. This is pretty much as fast as I could think about it.
I will take you through my thought process as I go along, which may make for an interesting piece. My thoughts are italicized.
After reading the question:
(Ok, I am going to have two equations, and probably something to do with divisibity, ok let's set up the equations).

Let the old books be o and new ones be n.
Then
1) 15n+10o = 125
(Hey I can reduce this one, less math work afterwards, here goes)
3n+2o = 25

(Ok, here's my first equation, let's write the other one)
2) Profit = 5n+2o.
(Oh, I can sub the first equation in this one to reduce it, awesome :))
Profit = 2n + (3n+2o) = 2n+25.

(Ok, profit is at least 25, it there a smaller option ? Damn, no. Well, Profit has to be an odd number since 2n is even, Is there an even number in the options? Damn, all are odd. What do I do?).

After about 10 seconds:

(Maybe I can use the original equation and divisibility stuff to do this one, let's see what happens?)
3n+2o = 25
3n = 25-2o
n= (25-2o)/3

(Hey I need 2n, let's multiply both sides by 2)
2n = 2*(25-2o)/3
(Hey the RHS has to be an integer, hence 25-2o must be a multiple of 3, let's check, o=1 will give me 23, that's not right. o=2 will give me, right, 21. 21 over 3 is 7, hence 2n becomes 14.)

Profit = 25+14 = 39.

(So 39 is not the answer, but wait, for other values of o, as o keeps increasing, 2n will keep decreasing, hence 39 is the highest value. Yes!. 41 is wrong. What a question!)


E is the right answer.

PS: Apparently, I call myself "Hey" and use the word "Ok" a lot in my head. ;)
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by tutorphd » Mon Jul 30, 2012 7:16 pm
This question definitely takes long time because you have to trace many possibilities. I classify it as "equations in integers".

x = new books; y = used books

sales: 15x+10y=125, looking for non-negative integer solutions for x and y
profit: 5x+2y


Always simplify equations in integers to lowest possible integers as coefficients. In this problem that is accomplished by dividing the sales equation by 5:

3x + 2y = 25

Focus on the variable with the largest coefficient (in this case x) and see what values are allowed for it: x=0 to 8. You focus on the variable with the largest coefficient because it depletes the sum (25) with fewer possible values.

Culling the allowed values by using divisibility: 2y is divisible by 2 -> 25-3x should be divisible by 2 i.e. should be even -> 3x should be odd -> x should be odd. If you don't notice that, you will have to try for twice as many possible values of x and you will notice that the even x do not give an integer y.

So x could be 1, 3, 5, 7. Solve for the corresponding y and find the corresponding profit=5x+2y:

x y 5x+2y
1 11 27
3 8 31
5 5 35
7 2 39

The only answer left is 41.
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