An antiques dealer purchased three cabinets at three distinct costs last year and resold all three of those cabinets for three distinct prices this year. If the median price was received for the cabinet that had cost the median amount, and the antiques dealer made a 10% profit on that cabinet, did the dealer make more than a 10% profit margin on any one of the three cabinet sales?
(1) One of the cabinets sold for a price less than its original cost.
(2) The cabinet that sold for the lowest price was the one that cost the antiques dealer the most to
Let C1,C2 and C3 be the Cost Price (CP increasing in that order) and S1,S2 and S3 are the Selling Price (SP increasing in that order)
Now given, (S2-C2)/C2*100= 10 % Profit Margin
From St1, We can have 2 cases i.e S1<C1 or S3<C3. But we don't know which one and we can only conclude on at least on one of these sale there will be a loss
From St 2, we get S1 was the Selling price of One Antique (Let it be Antique no 1) and C3 was the Cost Price
Now Imagine a case where the Selling price is same in all cases i.e S1=S2=S3=S
therefore, we have
S1</ S2</ S3---->S1-C3<S2-C2< S3-C1---> S-C3<S-C2<S-C1---->Divide Eqn by C1 we get
(S-C3)/C1<(S-C2)/C1<(S-C1)/C1-----> (S-C3)/C1 <(S-C2)/C1<(S/C1-1)
Now (S-C2)/C2*100=10% this implies (S-C2)/C1*100> (S-C2)/C2*100 (Because C2>C1)
and therefore (S/C1-1) is greater than 10% and Hence Ans B
It looked tedious but wanted to arrive at the solution. I have taken the Selling price same for all is to get at minimum values of S and if at that minimum value if we can get a profit margin more than 10% then it will be also applicable for S1<S2<S3
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