A shopkeeper marks up a pen at a certain percentage above its cost price. If the marked price of the pen is Rs. $Y$, find the value of $Y$.

Given the following statements:

**Statement I:** $Y$ is marked up 80\% above the cost price.

**Statement II:** The cost price of the pen is Rs. $X$. If the shopkeeper allows a discount of 4\% on the marked price, he makes a profit of 8\%.

**Statement III:** The cost price of the pen is Rs. $P$. If the shopkeeper allows a discount of $22\frac{2}{9}\%$ on the marked price, he makes a profit of Rs. 70.

Which of the following options is correct regarding the sufficiency of the statements to find $Y$?

Question

A shopkeeper marks up a pen at a certain percentage above its cost price. If the marked price of the pen is Rs. $Y$, find the value of $Y$.

Given the following statements:

**Statement I:** $Y$ is marked up 80\% above the cost price.

**Statement II:** The cost price of the pen is Rs. $X$. If the shopkeeper allows a discount of 4\% on the marked price, he makes a profit of 8\%.

**Statement III:** The cost price of the pen is Rs. $P$. If the shopkeeper allows a discount of $22\frac{2}{9}\%$ on the marked price, he makes a profit of Rs. 70.

Which of the following options is correct regarding the sufficiency of the statements to find $Y$?

Accepted Answer

**Solution:**

1. **From Statement I:**
   - We know the marked price $Y$ is 80\% above the cost price.
   - However, the cost price is unknown, so we cannot find the exact value of $Y$.

2. **From Statement II:**
   - Cost price = Rs. $X$.
   - Profit = 8\%, so selling price = $1.08X$.
   - Discount on marked price = 4\%, so selling price = 96\% of marked price.

Therefore,
   $$0.96Y = 1.08X \ \Rightarrow Y = \frac{1.08X}{0.96} = 1.125X$$

- Also, when sold at marked price $Y$, profit = Rs. 28:
   $$Y - X = 28$$

Substitute $Y = 1.125X$:
   $$1.125X - X = 28 \ 0.125X = 28 \ X = 224$$

Then,
   $$Y = 1.125 	imes 224 = 252$$

So, Statement II alone is sufficient to find $Y$.

3. **From Statement III:**
   - Cost price = Rs. $P$.
   - Discount = $22\frac{2}{9}\% = \frac{2}{9} 	imes 100\% = 22.22\%$.
   - Profit = Rs. 70, so selling price = $P + 70$.

Selling price after discount:
   $$	ext{Selling Price} = 	ext{Marked Price} 	imes (1 - 0.2222) = 0.7778Y$$

Given,
   $$0.7778Y = P + 70 \ \Rightarrow Y = \frac{P + 70}{0.7778} = \frac{9}{7}(P + 70)$$

4. **Using Statements I and III together:**
   - From I, $Y = P 	imes 1.8$ (since 80\% markup)
   - From III, $Y = \frac{9}{7}(P + 70)$

Equate:
   $$1.8P = \frac{9}{7}(P + 70)$$

Multiply both sides by 7:
   $$12.6P = 9P + 630 \ 12.6P - 9P = 630 \ 3.6P = 630 \ P = 175$$

Then,
   $$Y = 1.8 	imes 175 = 315$$

Thus, either Statement II alone or Statements I and III together are sufficient to find $Y$.

**Answer:** Either statement (II) alone or statements (I) and (III) together are sufficient to answer the question.

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