If a shopkeeper first raises the price of jewelery by x% then he decreases the…
2024
If a shopkeeper first raises the price of jewelery by x% then he decreases the new price by x%. After one such up down cycle, the price of a jewellery decreased by Rs. 1200. After a second up down cycle the jewellery was sold for Rs. 27648. What was the original price of jewellery ?
- A.
Rs. 24000
- B.
Rs. 36000
- C.
Rs. 44000
- D.
Rs. 30000
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Correct answer: D
Answer: Rs. 30000
Let the initial price be a. After increasing by x% and then decreasing by x%, the net multiplier is k = (1 + x/100)(1 - x/100) = 1 - x^2/10000.
After the first up-down cycle the price becomes a·k. The decrease from the original price is a - a·k = a(1 - k) which is given as 1200, so 1 - k = 1200/a and hence k = 1 - 1200/a.
After the second cycle the price is a·k^2 = 27648. Substitute k to get the equation a(1 - 1200/a)^2 = 27648.
Expand and simplify: (1 - 1200/a)^2 = 27648/a, so 1 - 2400/a + 1440000/a^2 = 27648/a. Multiply both sides by a^2 to obtain a^2 - 2400a + 1440000 = 27648a.
Bring all terms to one side: a^2 - 30048a + 1440000 = 0. Factor the quadratic: (a - 30000)(a - 48) = 0, so a = 30000 or a = 48.
Since the decrease after the first cycle is 1200, the original price must be greater than 1200, so a = 48 is not valid. Therefore the original price is Rs. 30000.