On selling each of the two radios for 5000, a person neither gained nor lost. If he had sold one radio at 25% gain, then at what per cent loss did she sell the 2nd radio?

Options

A. 16 `2/3` %
B. 18 `2/9` %
C. 0.25
D. 26 `2/3` %

Answer

To solve the problem step-by-step, we will follow these steps:

### Step 1: Understand the Selling Price and Gain
The selling price (SP) of each radio is given as 5000. The person neither gained nor lost on selling both radios, which means the total cost price (CP) of both radios is equal to the total selling price.

### Step 2: Calculate the Cost Price of the First Radio
If the first radio is sold at a 25% gain, we can use the formula for selling price:
\[ SP = CP + \text{Gain} \]
Where Gain is given by:
\[ \text{Gain} = \frac{25}{100} \times CP \]
Thus, we can express SP as:
\[ SP = CP + \frac{25}{100} \times CP = \frac{125}{100} \times CP \]
Given that SP = 5000, we can set up the equation:
\[ 5000 = \frac{125}{100} \times CP \]

### Step 3: Solve for the Cost Price of the First Radio
Rearranging the equation to find CP:
\[ CP = 5000 \times \frac{100}{125} \]
Calculating this gives:
\[ CP = 5000 \times 0.8 = 4000 \]
So, the cost price of the first radio is 4000.

### Step 4: Total Cost Price of Both Radios
Since the person sold both radios for a total of 10000 (5000 + 5000), and the first radio cost 4000, we can find the cost price of the second radio:
\[ \text{Total CP} = 4000 + CP_2 = 10000 \]
This implies:
\[ CP_2 = 10000 - 4000 = 6000 \]

### Step 5: Calculate the Loss on the Second Radio
The selling price of the second radio is also 5000. To find the loss, we use the formula:
\[ \text{Loss} = CP - SP \]
Thus:
\[ \text{Loss} = 6000 - 5000 = 1000 \]

### Step 6: Calculate the Percentage Loss
To find the percentage loss, we use the formula:
\[ \text{Percentage Loss} = \left( \frac{\text{Loss}}{CP} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Loss} = \left( \frac{1000}{6000} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage Loss} = \frac{1000 \times 100}{6000} = \frac{100000}{6000} = \frac{1000}{60} = \frac{100}{6} \approx 16.67\% \]

### Final Answer
Thus, the percentage loss on the second radio is approximately **16.67%** or **16 2/3%**.

---