A man sold two bicycles for Rs. 6000 each, gaining 20% on the one and losing 20% on the other. Find his gain or loss per cent on the whole transaction.

Answer

To solve the problem step-by-step, we will calculate the cost price (CP) of each bicycle, determine the total cost price and selling price, and finally find the overall gain or loss percentage.

### Step 1: Calculate the Cost Price of the First Bicycle (CP1)
Given:
- Selling Price (SP1) = Rs. 6000
- Gain = 20%

Let the Cost Price of the first bicycle be CP1 = Rs. X.

Using the formula for gain:
\[ \text{Gain} = \text{SP} - \text{CP} \]
\[ \text{Gain} = 20\% \text{ of CP1} = \frac{20}{100} \times X = \frac{X}{5} \]

So, we have:
\[ \text{SP1} = \text{CP1} + \text{Gain} \]
\[ 6000 = X + \frac{X}{5} \]

To solve for X, we can express it as:
\[ 6000 = \frac{5X + X}{5} \]
\[ 6000 = \frac{6X}{5} \]

Now, multiplying both sides by 5:
\[ 30000 = 6X \]
\[ X = 5000 \]

Thus, the Cost Price of the first bicycle (CP1) is Rs. 5000.

### Step 2: Calculate the Cost Price of the Second Bicycle (CP2)
Given:
- Selling Price (SP2) = Rs. 6000
- Loss = 20%

Let the Cost Price of the second bicycle be CP2 = Rs. Y.

Using the formula for loss:
\[ \text{Loss} = \text{CP} - \text{SP} \]
\[ \text{Loss} = 20\% \text{ of CP2} = \frac{20}{100} \times Y = \frac{Y}{5} \]

So, we have:
\[ \text{SP2} = \text{CP2} - \text{Loss} \]
\[ 6000 = Y - \frac{Y}{5} \]

To solve for Y, we can express it as:
\[ 6000 = \frac{5Y - Y}{5} \]
\[ 6000 = \frac{4Y}{5} \]

Now, multiplying both sides by 5:
\[ 30000 = 4Y \]
\[ Y = 7500 \]

Thus, the Cost Price of the second bicycle (CP2) is Rs. 7500.

### Step 3: Calculate Total Cost Price (CP) and Total Selling Price (SP)
Now, we can find the total cost price and total selling price:
\[ \text{Total CP} = CP1 + CP2 = 5000 + 7500 = 12500 \]
\[ \text{Total SP} = SP1 + SP2 = 6000 + 6000 = 12000 \]

### Step 4: Determine Overall Gain or Loss
Now, we can determine if there is an overall gain or loss:
\[ \text{Loss} = \text{Total CP} - \text{Total SP} = 12500 - 12000 = 500 \]

### Step 5: Calculate Loss Percentage
To find the loss percentage, we use the formula:
\[ \text{Loss Percentage} = \left( \frac{\text{Loss}}{\text{Total CP}} \right) \times 100 \]
\[ \text{Loss Percentage} = \left( \frac{500}{12500} \right) \times 100 = 4\% \]

### Final Answer
The man incurred a loss of 4% on the whole transaction.

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