A shopkeeper sold a TV set for Rs. 17940, with a discount of 8% and gained 19.6%. If no discount is allowed, then what will be his gain per cent ?

Options

A. `25%`
B. `26.4%`
C. `24.8%`
D. `30%`

Answer

To solve the problem step by step, we need to find the gain percentage if no discount is allowed. Let's break it down:

### Step 1: Understand the given information
- Selling Price (SP) = Rs. 17,940
- Discount = 8%
- Gain (Profit) = 19.6%

### Step 2: Calculate the Cost Price (CP)
To find the Cost Price (CP), we can use the relationship between Selling Price, Cost Price, and Profit Percentage.

The formula for Selling Price when profit is involved is:
\[ SP = CP + \text{Profit} \]
Where Profit can be expressed as:
\[ \text{Profit} = \frac{Profit \%}{100} \times CP \]

Given that the profit percentage is 19.6%, we can express this as:
\[ SP = CP + \frac{19.6}{100} \times CP \]
\[ SP = CP \left(1 + \frac{19.6}{100}\right) \]
\[ SP = CP \left(\frac{119.6}{100}\right) \]

Now, we can rearrange this to find CP:
\[ CP = \frac{SP \times 100}{119.6} \]

### Step 3: Substitute the Selling Price
Now, substitute the Selling Price into the equation:
\[ CP = \frac{17940 \times 100}{119.6} \]

### Step 4: Calculate CP
Calculating this gives:
\[ CP = \frac{1794000}{119.6} \approx 14997.49 \]

### Step 5: Calculate the Market Price (MP)
The Market Price (MP) can be calculated using the discount percentage. The formula for Selling Price with discount is:
\[ SP = MP \times \left(1 - \frac{Discount \%}{100}\right) \]

Rearranging gives us:
\[ MP = \frac{SP}{1 - \frac{Discount \%}{100}} \]
Substituting the discount percentage:
\[ MP = \frac{17940}{1 - 0.08} \]
\[ MP = \frac{17940}{0.92} \]

### Step 6: Calculate MP
Calculating this gives:
\[ MP \approx 19500 \]

### Step 7: Calculate Gain Percentage if no discount is allowed
If no discount is allowed, the gain percentage can be calculated using the formula:
\[ \text{Gain \%} = \frac{MP - CP}{CP} \times 100 \]

Substituting the values:
\[ \text{Gain \%} = \frac{19500 - 14997.49}{14997.49} \times 100 \]

### Step 8: Calculate Gain Percentage
Calculating this gives:
\[ \text{Gain \%} \approx \frac{4502.51}{14997.49} \times 100 \approx 30.01\% \]

### Final Answer
The gain percentage if no discount is allowed is approximately **30%**.

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