A man sold two radios for ? 2000 each. On one he gains 16% and on the other he losses 16%. Find his gain or loss per cent in the whole transaction.

Answer

To solve the problem step by step, we will find the cost price of each radio, calculate the total cost price, and then determine the overall gain or loss percentage.

### Step 1: Find the Cost Price of the First Radio
Given:
- Selling Price (SP1) = ₹2000
- Gain on the first radio = 16%

Let the Cost Price of the first radio be CP1.

Using the formula for selling price with profit:
\[ SP = CP + \text{Gain} \]
\[ SP = CP + \left(\frac{\text{Gain\%}}{100} \times CP\right) \]
\[ 2000 = CP1 + \left(\frac{16}{100} \times CP1\right) \]

This can be rewritten as:
\[ 2000 = CP1 \left(1 + \frac{16}{100}\right) \]
\[ 2000 = CP1 \left(\frac{116}{100}\right) \]

Now, solving for CP1:
\[ CP1 = 2000 \times \frac{100}{116} \]
\[ CP1 = 2000 \times \frac{100}{116} \approx 1724.14 \]

### Step 2: Find the Cost Price of the Second Radio
Given:
- Selling Price (SP2) = ₹2000
- Loss on the second radio = 16%

Let the Cost Price of the second radio be CP2.

Using the formula for selling price with loss:
\[ SP = CP - \text{Loss} \]
\[ SP = CP - \left(\frac{\text{Loss\%}}{100} \times CP\right) \]
\[ 2000 = CP2 - \left(\frac{16}{100} \times CP2\right) \]

This can be rewritten as:
\[ 2000 = CP2 \left(1 - \frac{16}{100}\right) \]
\[ 2000 = CP2 \left(\frac{84}{100}\right) \]

Now, solving for CP2:
\[ CP2 = 2000 \times \frac{100}{84} \]
\[ CP2 = 2000 \times \frac{100}{84} \approx 2380.95 \]

### Step 3: Calculate Total Cost Price
Now, we find the total cost price (CP) of both radios:
\[ \text{Total CP} = CP1 + CP2 \]
\[ \text{Total CP} = 1724.14 + 2380.95 \]
\[ \text{Total CP} \approx 4105.09 \]

### Step 4: Calculate Total Selling Price
The total selling price (SP) of both radios is:
\[ \text{Total SP} = SP1 + SP2 \]
\[ \text{Total SP} = 2000 + 2000 = 4000 \]

### Step 5: Determine Gain or Loss
Now we can determine if there is a gain or loss:
\[ \text{Loss} = \text{Total CP} - \text{Total SP} \]
\[ \text{Loss} = 4105.09 - 4000 \]
\[ \text{Loss} \approx 105.09 \]

### Step 6: Calculate Loss Percentage
Finally, we calculate the loss percentage:
\[ \text{Loss\%} = \left(\frac{\text{Loss}}{\text{Total CP}}\right) \times 100 \]
\[ \text{Loss\%} = \left(\frac{105.09}{4105.09}\right) \times 100 \]
\[ \text{Loss\%} \approx 2.56\% \]

### Final Answer
The man incurred a loss of approximately **2.56%** in the whole transaction.

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