An item with a cost price of Rs. 1650 is sold at a certain discount on a fixed marked price to earn a profit of 20% on the cost price. If the discount was doubled, the profit would have been Rs.
110
The rate of discount, in percentage, at which the profit percentage would be equal to the rate of discount, is nearest to
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Official Correct Answer: C. The question says that an item with a cost price of Rs. 1650 is sold at a certain discount on a fixed marked price, earning a profit of 20% on the cost price. Then, if the discount was doubled, the profit would have been Rs. 110. We need to find the rate of discount where the profit percentage equals the discount rate, and choose the nearest option from A to D.
First, let's break down the given information. The cost price (CP) is Rs. 1650. The selling price (SP) when the profit is 20% on CP would be CP plus 20% of CP. So, SP = 1650 + 0.2*1650. Calculate that: 0.2*1650 is 330, so SP is 1650 + 330 = 1980. So, the selling price is Rs. 1980.
Now, this selling price is after a certain discount on the marked price (MP). Let the discount percentage as 'd'. So, the selling price can also be expressed as MP minus d% of MP. In other words, SP = MP*(1 - d/100).
But wait, we don't know the marked price yet. Hmm, maybe we can express MP in terms of SP and d. Let's rearrange the equation: MP = SP / (1 - d/100). Since SP is 1980, MP = 1980 / (1 - d/100).
Now, the second part of the question says that if the discount was doubled, the profit would have been Rs. 110. So, if the discount becomes 2d%, then the new selling price (SP2) would be MP*(1 - 2d/100). The profit in this case would be SP2 - CP, which is given as 110.
So, SP2 = CP + 110 = 1650 + 110 = 1760. Therefore, SP2 = 1760.
But SP2 is also equal to MP*(1 - 2d/100). So, we have two expressions for MP:
1. MP = 1980 / (1 - d/100)
2. MP = 1760 / (1 - 2d/100)
Since both equal MP, we can set them equal to each other:
1980 / (1 - d/100) = 1760 / (1 - 2d/100)
Now, let's solve for d. Cross-multiplying gives:
1980*(1 - 2d/100) = 1760*(1 - d/100)
Expanding both sides:
1980 - (1980*2d)/100 = 1760 - (1760*d)/100
Simplify the terms:
1980 - (3960d)/100 = 1760 - (1760d)/100
Let me convert the coefficients to make it easier:
1980 - 39.6d = 1760 - 17.6d
Now, let's bring like terms together. Subtract 1760 from both sides:
1980 - 1760 - 39.6d = -17.6d
Which simplifies to:
220 - 39.6d = -17.6d
Now, add 39.6d to both sides:
220 = 22d
So, d = 220 / 22 = 10.
Wait, that gives d = 10. But looking at the options, 10 isn't one of them. The options are 12, 14, 16, 18. Hmm, maybe I made a mistake somewhere.
Starting from the equation:
1980*(1 - 2d/100) = 1760*(1 - d/100)
Expanding:
1980 - (1980*2d)/100 = 1760 - (1760*d)/100
Which is:
1980 - 39.6d = 1760 - 17.6d
Subtract 1760:
220 - 39.6d = -17.6d
Add 39.6d:
220 = 22d
So, d = 10. Hmm, that's correct. But the options don't have 10. Maybe I misunderstood the problem.
Wait, the question asks for the rate of discount where the profit percentage equals the discount rate. So, initially, we found d = 10, but that's not the answer we're looking for. The answer is when profit percentage equals discount rate, which is a different scenario.
So, let me rephrase. We need to find a discount rate 'x' such that the profit percentage is equal to 'x'.
Let the discount rate as 'x'. So, the selling price when discount is x% is SP = MP*(1 - x/100). The profit is SP - CP, and the profit percentage is (SP - CP)/CP * 100. We need this profit percentage to be equal to x.
So, (SP - CP)/CP * 100 = x
But SP = MP*(1 - x/100), so:
[(MP*(1 - x/100) - CP)/CP] * 100 = x
We can express MP in terms of the original selling price. From the first part, when discount was d = 10%, MP = 1980 / (1 - 10/100) = 1980 / 0.9 = 2200.
So, MP is 2200. Now, substituting MP = 2200 into the equation:
[(2200*(1 - x/100) - 1650)/1650] * 100 = x
Simplify the numerator:
2200*(1 - x/100) - 1650 = 2200 - 22x - 1650 = 550 - 22x
So, the equation