psm12se 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. $39
E. $41
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 equation from the top equation 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.
If we're lucky, there will be only 1 odd answer choice, in which case we can save a lot of time.
When I check the answer choices, I can see that all of the answer choices are odd. Too bad
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.
Keep trying each answer choice, and you'll find that A, B, C and D are all possible.
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
