What per cent above cost price.should goods be marked for sale so that after allowing12 `1/2` %trade discount and 5% cash discount, a net gain of 33% may be earned?

Options

A. A) 45%
B. B) 40%
C. C) 50%
D. D) 60%

Answer

To solve the problem step by step, we need to determine the percentage above the cost price (CP) at which goods should be marked for sale (marked price, MP) to achieve a net gain of 33% after allowing for trade and cash discounts.

### Step 1: Understand the Discounts
We have two discounts:
1. Trade discount = 12.5%
2. Cash discount = 5%

### Step 2: Calculate the Net Discount
To find the net discount percentage, we can use the formula for successive discounts:

If the first discount is \(d_1\) and the second discount is \(d_2\), the net discount \(D\) can be calculated as:
\[ D = d_1 + d_2 - \left(\frac{d_1 \times d_2}{100}\right) \]

Substituting the values:
\[ D = 12.5 + 5 - \left(\frac{12.5 \times 5}{100}\right) \]
\[ D = 12.5 + 5 - 0.625 = 16.875\% \]

### Step 3: Calculate the Selling Price (SP) and Cost Price (CP) Relationship
We want to find the relationship between the marked price (MP) and the cost price (CP) after accounting for the net discount and the desired profit.

The formula to relate CP, MP, and profit percentage is:
\[ \frac{CP}{MP} = \frac{100 - D}{100 + P} \]
Where:
- \(D\) = net discount percentage
- \(P\) = profit percentage

Here, \(D = 16.875\%\) and \(P = 33\%\).

Substituting the values:
\[ \frac{CP}{MP} = \frac{100 - 16.875}{100 + 33} \]
\[ \frac{CP}{MP} = \frac{83.125}{133} \]

### Step 4: Simplify the Ratio
To simplify the ratio:
\[ \frac{CP}{MP} = \frac{83.125}{133} \approx \frac{5}{8} \]

### Step 5: Calculate the Markup Percentage
The markup percentage can be calculated using the formula:
\[ \text{Markup Percentage} = \left(\frac{MP - CP}{CP}\right) \times 100 \]

From the ratio \( \frac{CP}{MP} = \frac{5}{8} \), we can express:
- Let \(CP = 5x\) and \(MP = 8x\).

Now substituting in the markup formula:
\[ \text{Markup Percentage} = \left(\frac{8x - 5x}{5x}\right) \times 100 = \left(\frac{3x}{5x}\right) \times 100 = 60\% \]

### Conclusion
The goods should be marked at **60% above the cost price** to achieve the desired profit after discounts.