By how much %, a shopkeeper must mark his goods above its CP so as by giving 13% discount he may gain 45%?

Options

A. `50%`
B. `60%`
C. `66.66%`
D. `63.36%`

Answer

To solve the problem step by step, we need to determine by what percentage the shopkeeper must mark his goods above the cost price (CP) to achieve a 45% profit after giving a 13% discount.

### Step-by-Step Solution:

1. **Let the Cost Price (CP) be Rs 100.**
- This simplifies calculations, as we can work with percentages directly.

**Hint:** Starting with a base value (like Rs 100) makes it easier to calculate percentages.

2. **Calculate the Selling Price (SP) after a 13% discount.**
- The discount is 13% of the marked price (MP).
- The Selling Price (SP) can be calculated as:
\[ SP = MP - \text{Discount} \]
- Since we are assuming the Marked Price (MP) is Rs 100, the Selling Price becomes:
\[ SP = 100 - 13 = 87 \text{ Rs} \]

**Hint:** Remember that the Selling Price is the Marked Price minus the discount.

3. **Determine the desired Selling Price for a 45% profit.**
- To find the Selling Price that gives a 45% profit on the Cost Price (CP), we calculate:
\[ SP = CP + \text{Profit} \]
- The profit is 45% of the CP (which we assumed to be Rs 100):
\[ \text{Profit} = 45\% \text{ of } 100 = 45 \text{ Rs} \]
- Therefore, the Selling Price for a 45% profit is:
\[ SP = 100 + 45 = 145 \text{ Rs} \]

**Hint:** The Selling Price can also be calculated as CP multiplied by (1 + Profit Percentage).

4. **Set the Selling Price from step 2 equal to the Selling Price from step 3.**
- We know from step 2 that:
\[ SP = 87 \text{ Rs} \]
- And from step 3:
\[ SP = 145 \text{ Rs} \]
- We need to find the Marked Price (MP) that will allow the Selling Price to be Rs 145 after a 13% discount.

5. **Calculate the Marked Price (MP).**
- The Selling Price (SP) is related to the Marked Price (MP) by the formula:
\[ SP = MP - \text{Discount} \]
- Rearranging gives:
\[ MP = SP + \text{Discount} \]
- The discount is 13% of the Marked Price, so we can express it as:
\[ \text{Discount} = 0.13 \times MP \]
- Thus:
\[ SP = MP - 0.13 \times MP = 0.87 \times MP \]
- Setting this equal to the Selling Price for profit:
\[ 0.87 \times MP = 145 \]
- Solving for MP gives:
\[ MP = \frac{145}{0.87} \approx 166.67 \text{ Rs} \]

**Hint:** To find the Marked Price, you can rearrange the formula to isolate MP.

6. **Calculate the percentage markup over the Cost Price (CP).**
- The markup is given by:
\[ \text{Markup} = MP - CP = 166.67 - 100 = 66.67 \text{ Rs} \]
- The percentage markup over the Cost Price is:
\[ \text{Percentage Markup} = \left(\frac{\text{Markup}}{CP}\right) \times 100 = \left(\frac{66.67}{100}\right) \times 100 = 66.67\% \]

**Hint:** The percentage markup is calculated as the difference between the Marked Price and Cost Price divided by the Cost Price, multiplied by 100.

### Final Answer:
The shopkeeper must mark his goods approximately **66.67%** above the Cost Price to achieve a 45% profit after giving a 13% discount.