Did the seller earn a profit?

Question

(1) The selling price of 20 items equals the cost of 15 items.

(2) The cost of 10 items is $55 greater than the selling price of 8 items.

Source: GMAT Math BIBLE, p.444, q.152

Explanation from the source:

Let c and s be the cost and the selling price, respectively, for the seller on each item.

From Statement (1) alone, we have that the selling price of 20 items equals the cost of 15 items.
Hence, we have 20s = 15c, or s = ( \frac{15}{20} )c =  \frac{3c}{4} .
Now, profit = selling price – cost = s – c =  \frac{3c}{4}  – c =  \frac{–c}{4} , a negative value (c is price and therefore is positive. Hence, –c/4 is negative).
A negative profit should be considered a loss. Hence, Statement (1) alone is sufficient to determine that the seller did not earn a profit.

From Statement (2) alone, we have that the cost of 10 items is $55 more than the selling price of 8 items.
The cost of 10 items is 10c and the selling price of 8 items is 8s.
Hence, we have 10c – 8s = 55.
Subtracting 2s from both sides yields 10c – 10s = 55 – 2s; 10(c – s) = 55 – 2s; c – s =  \frac{(55 – 2s)}{10} .
Now, if 55 – 2s is positive, c – s, which equals  \frac{(55 – 2s)}{10} , is positive. Here, Cost > Selling Price, and seller made a profit.
Otherwise, he did not make a profit. 55 – 2s is not positive if s < 55/2.
Hence, we have a double case and Statement (2) alone is not sufficient.

LevanKhukhunashvili · Answer

Honestly, I don't like the wording of the question, but who cares

1 stm - it's clear that our selling price is less than the cost, as selling 20 items will cover the costs only for 15 items. We will lose if we sell any. Suff 2 stm - 10*cost-8*price=55 We can manipulate a lot with two variables. Not suff IMO Ans: A

Manat · Answer

chetan2u  :-how is statement 2 not sufficient. Please explain, i can't find a scenario where it will lead to a profit for the seller.

MahmoudFawzy · Answer

hi  Manat

for statement 2:
 10c = 8p + 55 ,  c =   \frac{8p + 55}{10}

if  p\leq{27} , then  c\geq{27.1}  --> loss
if  p = 27.5 , then  c = 27.5  --> break even (when  p = c ,  10c = 8c + 55 ,  2c = 55 ,  c = p = 27.5 )
if  p\geq{28}  , then  c\leq{27.9}  --> profit

while in statement 1, it is locked ratio and c is always greater as long as we are dealing with positive numbers.
 20s = 15c ,  \frac{c}{s} = \frac{20}{15}  ---> (we can never assume that c = p as in statement 2)

mangamma · Answer

how statement 1 is sufficient?

neha283 · Answer

Statement 1 :
20s=15c

\frac{s}{c}= \frac{15}{20}

which means selling price is less than cost price, hence the seller went in loss. 
Question - "Did the seller earn a profit?" -- Answer - No
So, statement 1 is sufficient.

chetan2u · Answer

Hi  Manat 
The difference 55 holds the key..

If the cost of two items is more than 55, then profit..
Say each costs 50, so cost of 10 items is 50*10=500.
But 500-55=445 is the selling price of 8 items.. so each item is sold for 445/8=~55
So profit of 5 per piece.

If the cost of two items is less than 55, then loss..
Say each costs 10, so cost of 10 items is 10*10=100.
But 100-55=45 is the selling price of 8 items.. so each item is sold for 45/8=~5.5
So loss of 5 per piece.

tannumunu · Answer

what if I solve the question with Option II in the following manner :-

let assume the S.P. of 8 items are $X
the S.P. of 1 item is $X/8

C.P of 10 items are $55
C.P of 1 items $5.5

now based on 1 item (S.P - C.P) will give us a negative value which implies the seller did loss by selling the product.

Please let me know where I am thinking wrong with this procedure.

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Did the seller earn a profit?

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