Quote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?
A) 11
B) 12
C) 13
D) 14
E) 15
We can PLUG IN THE ANSWERS, which represent the value of n.
When the correct answer choice is plugged in, the total sold will be less than n.
D: n=14
Since Mark sold 10 boxes less than n, the number sold by Mark = 14-10 = 4.
Since Ann sold 2 boxes less than n, the number sold by Ann = 14-2 = 12.
Total sold = 4+12 = 16.
Here, the total sold is GREATER THAN n.
Eliminate D.
B: n=12
Since Mark sold 10 boxes less than n, the number sold by Mark = 12-10 = 2.
Since Ann sold 2 boxes less than n, the number sold by Ann = 12-2 = 10.
Total sold = 2+10 = 12.
Here, the total sold is EQUAL TO n.
Eliminate B.
Notice the trend:
n=14 yields a sales volume GREATER THAN n.
n=12 yields a sales volume EQUAL TO n.
Implication:
A smaller value for n is required to yield a sales volume LESS THAN n.
Algebraically:
Mark's sales = n-10.
Ann's sales = n-2.
Since the total sold must be less than n, we get:
(n-10) + (n-2) < n
2n-12 < n
n < 12.
NandishSS wrote:
Here I got confused by If Mark and Ann have each sold at least one box of cookies.
Can you help to elaborate on how this statement affects as whole?
As shown in my solution, n<12.
Without the statement above, a test-taker might deduce two possible answers: n=10 or n=11.
The statement above renders n=10 inviable.
If n=10, then the number of boxes sold by Mark = n-10 = 10-10 = 0.
Not possible, given the condition that Mark sells at least one box.
Thus -- because of the condition that Mark sells at least one box -- the only viable solution is n=11.
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