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25 Mar 2019, 01:09
Kudos
Bookmarks
A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
76% (01:55) correct
24%
(02:43)
wrong
based on 54
sessions
History
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Not Attempted Yet
Oscar buys 13 pencils and 3 erasers for 1.00. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?
(A) 10
(B) 12
(C) 15
(D) 18
(E) 20
(A) 10
(B) 12
(C) 15
(D) 18
(E) 20
Archit3110
Major Poster
25 Mar 2019, 01:26
Kudos
Bookmarks
Bunuel
given
13x+3y=100
and x>y
max value x = 7 so y = 3
x+y= 10
IMO A
27 Mar 2019, 18:53
Kudos
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Bunuel
Letting p = the number of pencils purchased and e = the number of erasers purchased, we can create the equation:
13p + 3e = 100
3e = 100 - 13p
e = (100 - 13p)/3
In order for (100 - 13p) to be a multiple of 3, we need to have p = 1, 4, 7, etc.
If p = 1, then e = 87/3 = 29 (but this is not possible since a pencil costs more than an eraser).
If p = 4, then e = 48/3 = 16 (again this is not possible).
If p = 7, then e = 9/3 = 3 (this is possible).
If p = 10, then e = -30/3 = -10 (this is not possible since e can’t be negative, so we can stop here).
Thus, the total cost of one pencil and one eraser is 7 + 3 = 10 cents.
Alternate Solution:
Letting p denote the cost of a pencil (in cents) and e denote the cost of an eraser (in cents), we can create the equation:
13p + 3e = 100
If we let n be the cost of one pencil and one eraser, then p + e = n or, equivalently, 3p + 3e = 3n. Let’s subtract this equation from the equation we formed earlier:
10 p = 100 - 3n
p = 10 - 3n/10
Since the cost of a pencil must be a whole number, in cents, and since 3 is not divisible by 10, we see that n (the cost of one pencil and one eraser) must be a multiple of 10. We can eliminate all answer choices except A and E.
If n = 20, then p = 10 - (3*20)/10 = 10 - 6 = 4 cents and an eraser is 20 - 4 = 16 cents, which is not possible because we are given that the cost of a pencil is greater than the cost of an eraser.
We know at this point that the answer is A, but let’s verify:
If n = 10, then p = 10 - (3*10)/10 = 10 - 3 = 7 and e = 10 - 7 = 3. So, the cost of one pencil is 7 cents, and the cost of one eraser is 3 cents, which agrees with the information given in the question.
Answer: A
