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13 May 2009 14:34:53 IST

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Three different percentage discounts are offered by a shopkeeper to 3 different customers on an item

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Three different percentage discounts are offered by a shopkeeper to 3 different customers on an item. Shopkeeper marked the price of the item 25% above the cost price. If the prices at which 3 customers buys these items are in A.P. and the discount percentages offered are natural numbers, then in how many ways shopkeeper can do this so that he does not bear loss from any of these customers?OPTIONS 1) 150 2) 90 3) 360 4) 270 5) None of these

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RyuAmakusa

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14 May 2009 00:49:22 IST

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$\text{Let x be the actual cost of the object so he sets the price at }\frac{5x}{4}\\ \text{let the profit percentages be l,m,n and the customers buy it at a,b,c respectively then a,b,c are in AP and }\\ a= \frac{5x}{4}*(1-\frac{l}{100})\\ b= \frac{5x}{4}*(1-\frac{m}{100})\\ c= \frac{5x}{4}*(1-\frac{n}{100})\\ \text{we can see that there is a linear relation between (a,l) ; (b,m) ; (c,n)} \\ \text{and this relation is same so l,m,n are also in AP.Now there should be no loss so }\\ a \ge x \text{ which implies } l \le 20 \text{ since l is any variable the same condition applies for} \\ \text{m and n so we have to select 3 int in [1,20] such that they are in AP let the ints be }\\ l,l+d,l+2d \text{ now } d \le \frac{(20-l) }{2}\\l=1 \; d={1.....9}\\l=2 \; d={1.....9}\\l=3 \; d={1.....8}\\l=4 \; d={1.....8}\\ .\\ .\\ .\\l=17 \; d={1}\\l=18 \; d={1}\\ \text{so the ans is 90 ie.(2)}$

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