Question 1081185: Amanda bought erasers for $90. Then she sold it all with a profit of $2 for each eraser. If after selling the erasers, she will buy with additional 15 erasers, what is the cost of each eraser?
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Amanda bought erasers for $90. Then she sold it all with a profit of $2 for each eraser. If after selling the erasers, she buys an additional 15 erasers, what is the cost of each eraser?
let x = number of erasers.
let c = cost of each eraser.
xc = 90
this means that the number of erasers bought * the cost of each eraser = 90 dollars.
all of the erasers were sold at a profit of 2 dollars each.
if you let s = the total selling price, then you get:
s = 90 + 2x
with the selling price, she was able to buy 15 more erasers.
since the cost of each eraser is the same, this means that:
s = 15c
since s is also equal to 90 + 2x, this means that:
15c = 90 + 2x
since xc = 90, you can solve for x to get x = 90/c
replce 2x in the equation of 15c = 90 + 2x and you get:
15c = 90 + 2 * (90/c)
simplify to get:
15c = 90 + 180 / c
multiply both sides of this equation by c and you get:
15c^2 = 90c + 180
subtract 90c and 180 from both sides of the equation and you get:
15c^2 - 90c - 180 = 0
solve for c using the quadratic formula (or a calculator as i did) to get:
c = 7.5825756949558 or c = -1.5825756949558
since c has to be positive, then c has to be equal to 7.5825756949558
that means the cost of each eraser was $7.5825756949558.
since xc = 90, then x has to be equal to 90 / 7.5825756949558 = 11.86931771
that means that she bought 11.86931771 erasers at a cost of 7.5825756949558 each.
if she sold all of them at a profit of 2 dollars each, that means that she sold 11.86931771 erasers for 9.5825756949558 for a total selling price of 113.7386354 dollars.
with that money, she bought 15 additional erasers.
the cost of each eraser was therefore 113.7386354 / 15 = 7.582575695 each.
that's your answer.
the cost of each eraser was $7.582575695.
the number are not very pretty but the method is correct.
i tested the method by creating a problem that could be solved with nice numbers.
i assumed the cost of each eraser was 3 dollars each.
since the total cost was 90 dollars, then she originally bought 90/3 = 30 erasers.
so my assumption was that she bought 30 erasers at a cost of 3 dollars each.
now, if she sold those erasers for a profit of 2 dollars each, then she made a total of 30 * 5 = 150 dollars.
if you take 150 dollars and divide it by 3, you get that she was able to buy 50 erasers with the money she made by selling the original 30 erasers at 5 dollars each.
my improvised problem would therefore state:
Amanda bought erasers for $90. Then she sold it all with a profit of $2 for each eraser. If after selling the erasers, she buys an additional 50 erasers, what is the cost of each eraser?
i then proceeded to solve this problem the same way i solved your original problem.
xc = 90
s = 90 + 2x
50c = 90 + 2x
50c = 90 + 180/c
multiply both sides by c to get:
50c^2 = 90c + 180
subtract 90c and 180 from both sides of the equation to get:
50c^2 - 90c - 180 = 0
solve this quadratic equation to get:
c = 3 or c = -1.2
c has to be equal to 3.
when c = 3, xc = 90 becomes 3x = 90 which gets you x = 30
if you sell 30 at 5 dollars apiece, you make 150 dollars.
if you buy 50 additional erasers, then the cost of each eraser is 150/50 = 3.
same method get me an answer that looks a lot prettier than the answer from your original problem.
i believe the method is correct, therefore i believe the numbers you were given in the original problem were not given with the intention of getting you a pretty answer.
if they were given you with the intention of getting an answer that made sense, the number of erasers you bought originally and the number you subsequently bought with the selling price should have been integers.
they weren't.
in my contrived example, they were, mainly because i set the problem up so that they would be integers.
this is what i think is happening with this problem.
i'm pretty sure i'm correct, but there is always the probability that i'm wrong.
if i'm correct, then what i said above is accurate.
if i'm wrong?
then it's anybody's guess as to what the correct solution needs to be.
i did a separate test with the possible number of erasers originally bought equal to from 1 to 90.
i calculated what the cost would have been for each and then raised the cost by 2 dollars and then determined how many additional eraser i could have bought.
i came up with the following possible integer solutions:
if the original cost was 6, then you bought 15 and could have bought an additional 20.
if the original cost was 3, then you bought 30 and could have bought an additional 50.
if the original cost was 2, then you bought 45 and could have bought an additional 90
if the original cost was 1.5, then you bought 60 and could have bought an additional 140.
if the original cost was 1.2, then you bought 785 and could have bought an additional 200
if the original cost was 1, then you bought 90 and could have bought an additional 270.
so, the only possible original number of erasers she could have bought that lead to an integer solution are:
15, 30, 45, 60, 75, 90
the only possible number of additional erasers shw could have bought that lead to an integer solution are:
20, 30, 45, 60, 75, 90.
my guess is that the original number of erasers bought is 15 and she could have bought another 20 erasers with the money she earned after selling them at 2 dollars over cost.
with 15 eraser originally bought, the cost per eraser was 6 dollars.
if she sold all 15 at a profit of 2 dollars each, then she received 15 * 8 = 120 dollars.
if you take the 120 dollars and divide them by the original cost of 6, then she was able to buy an additional 120/6 = 20 erasers.
that's my guess.
Answer by MathTherapy(10556)  (Show Source):
You can put this solution on YOUR website! Amanda bought erasers for $90. Then she sold it all with a profit of $2 for each eraser. If after selling the erasers, she will buy with additional 15 erasers, what is the cost of each eraser?
Assuming that the PROCEEDS from the sale were used to buy the 15 extra erasers, then cost of each eraser is: 
I find this to be a very weird problem, since the total cost of $90, divided by the cost of each eraser (1.582575695) DOES NOT GIVE an integer number of erasers purchased.
Regardless, the cost of each eraser is as stated above.
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