A dealer sold a radio at a loss of 2.5%. Had he sold it for Rs. 100 more, he would have gained `7 (1)/(2)`%. In order to gain `12 (1)/(2)%`, he should sell it for:(in Rs.)
(a)850
(b)925
(c)1080
(d)1125

Options

A. Rs. 850
B. Rs. 925
C. Rs. 1,080
D. Rs. 1,125

Answer

To solve the problem step by step, we will follow the information given in the question and derive the necessary calculations.

### Step 1: Define the Cost Price (CP)
Let the Cost Price (CP) of the radio be Rs. 1000. This is a convenient assumption to simplify calculations since we will later find the actual CP.

### Step 2: Calculate the Selling Price (SP) at a Loss of 2.5%
The dealer sells the radio at a loss of 2.5%. Therefore, the Selling Price (SP) can be calculated as:
\[ SP = CP - \text{Loss} = CP - (2.5\% \text{ of } CP) \]
\[ SP = 1000 - (0.025 \times 1000) = 1000 - 25 = Rs. 975 \]

### Step 3: Determine the New Selling Price with a Rs. 100 Increase
According to the problem, if the dealer sold the radio for Rs. 100 more, he would have gained 7.5%. Therefore, the new Selling Price (SP') would be:
\[ SP' = SP + 100 = 975 + 100 = Rs. 1075 \]

### Step 4: Calculate the Gain Percentage
The gain percentage when selling at Rs. 1075 is given as 7.5%. We can express this as:
\[ \text{Gain} = SP' - CP = 1075 - CP \]
The gain percentage formula is:
\[ \text{Gain Percentage} = \left(\frac{\text{Gain}}{CP}\right) \times 100 \]
Substituting the values we have:
\[ 7.5 = \left(\frac{1075 - CP}{CP}\right) \times 100 \]
Solving for CP:
\[ 7.5 \times CP = 1075 - CP \]
\[ 7.5CP + CP = 1075 \]
\[ 8.5CP = 1075 \]
\[ CP = \frac{1075}{8.5} = Rs. 126.47 \text{ (approximately)} \]

### Step 5: Calculate the Selling Price for 12.5% Gain
Now, to find the Selling Price (SP) for a gain of 12.5%, we first express 12.5% as a fraction:
\[ 12.5\% = \frac{25}{2}\% \]
The Selling Price (SP) for a gain of 12.5% can be calculated as:
\[ SP = CP + \text{Gain} = CP + \left(12.5\% \text{ of } CP\right) \]
\[ SP = CP + \left(\frac{25}{2} \times \frac{CP}{100}\right) \]
Substituting the value of CP:
\[ SP = 1000 + \left(\frac{25}{2} \times \frac{1000}{100}\right) \]
\[ SP = 1000 + 125 = Rs. 1125 \]

### Conclusion
Thus, to gain 12.5%, the dealer should sell the radio for Rs. 1125.

### Answer
The correct option is (d) 1125.