An article is marked up by P% above the cost price. If shopkeeper offers discount of 25% and still earns a profit of 35%. For earning a profit of 71%, how much discount should be offered by shopkeeper?
Question
An article is marked up by P% above the cost price. If shopkeeper offers discount of 25% and still earns a profit of 35%. For earning a profit of 71%, how much discount should be offered by shopkeeper?
Solution
Let's break down the problem step by step:
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Let's assume the cost price of the article is C.
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The article is marked up by P% above the cost price, so the selling price would be C + (P/100) * C = C(1 + P/100).
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The shopkeeper offers a discount of 25%, which means the selling price after the discount would be (1 - 25/100) * C(1 + P/100) = 0.75 * C(1 + P/100).
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The shopkeeper still earns a profit of 35%, which means the selling price after the discount should be 1.35 times the cost price: 0.75 * C(1 + P/100) = 1.35 * C.
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Now, let's solve the equation to find the value of P: 0.75 * C(1 + P/100) = 1.35 * C.
Simplifying the equation, we get: 0.75(1 + P/100) = 1.35.
Dividing both sides by 0.75, we get: 1 + P/100 = 1.35/0.75.
Subtracting 1 from both sides, we get: P/100 = (1.35/0.75) - 1.
Multiplying both sides by 100, we get: P = 100 * ((1.35/0.75) - 1).
Evaluating the expression, we get: P ≈ 80.
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So, the article is marked up by 80% above the cost price.
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Now, we need to find out how much discount should be offered by the shopkeeper to earn a profit of 71%.
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Let's assume the new discount as D%.
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The selling price after the new discount would be (1 - D/100) * C(1 + P/100).
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The shopkeeper wants to earn a profit of 71%, so the selling price after the new discount should be 1.71 times the cost price: (1 - D/100) * C(1 + P/100) = 1.71 * C.
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Now, let's solve the equation to find the value of D: (1 - D/100) * C(1 + P/100) = 1.71 * C.
Simplifying the equation, we get: (1 - D/100)(1 + P/100) = 1.71.
Dividing both sides by (1 + P/100), we get: 1 - D/100 = 1.71/(1 + P/100).
Subtracting 1 from both sides, we get: -D/100 = 1.71/(1 + P/100) - 1.
Multiplying both sides by -100, we get: D = -100 * (1.71/(1 + P/100) - 1).
Evaluating the expression, we get: D ≈ -100 * (1.71/(1 + 80/100) - 1).
Simplifying further, we get: D ≈ -100 * (1.71/(1 + 0.8) - 1).
Evaluating the expression, we get: D ≈ -100 * (1.71/1.8 - 1).
Simplifying further, we get: D ≈ -100 * (0.95 - 1).
Evaluating the expression, we get: D ≈ -100 * (-0.05).
Simplifying further, we get: D ≈ 5.
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So, the shopkeeper should offer a discount of approximately 5% to earn a profit of 71%.
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