Mr. Sinha sold two tape-recorders for Rs 990 each, gaining 10% on one and losing 10% on the other. Find his total loss or gain, as percent, on the whole transaction.

Answer

To solve the problem, we need to determine Mr. Sinha's total gain or loss from selling two tape recorders, one at a gain of 10% and the other at a loss of 10%. Let's break it down step by step.

### Step 1: Determine the Selling Price (SP) and Gain for the First Tape Recorder
- Selling Price (SP1) of the first tape recorder = Rs 990
- Gain percentage on the first tape recorder = 10%

Using the formula for Selling Price:
\[ SP = CP + \text{Gain} \]
Where Gain = Gain Percentage × Cost Price (CP).

Let the Cost Price of the first tape recorder be \( CP_1 \).
\[ SP_1 = CP_1 + (0.1 \times CP_1) \]
\[ 990 = CP_1 + 0.1 \times CP_1 \]
\[ 990 = 1.1 \times CP_1 \]

Now, solving for \( CP_1 \):
\[ CP_1 = \frac{990}{1.1} = 900 \]

### Step 2: Calculate the Gain on the First Tape Recorder
Gain on the first tape recorder:
\[ \text{Gain} = SP_1 - CP_1 = 990 - 900 = 90 \]

### Step 3: Determine the Selling Price (SP) and Loss for the Second Tape Recorder
- Selling Price (SP2) of the second tape recorder = Rs 990
- Loss percentage on the second tape recorder = 10%

Using the formula for Selling Price with Loss:
\[ SP = CP - \text{Loss} \]
Where Loss = Loss Percentage × Cost Price (CP).

Let the Cost Price of the second tape recorder be \( CP_2 \).
\[ SP_2 = CP_2 - (0.1 \times CP_2) \]
\[ 990 = CP_2 - 0.1 \times CP_2 \]
\[ 990 = 0.9 \times CP_2 \]

Now, solving for \( CP_2 \):
\[ CP_2 = \frac{990}{0.9} = 1100 \]

### Step 4: Calculate the Loss on the Second Tape Recorder
Loss on the second tape recorder:
\[ \text{Loss} = CP_2 - SP_2 = 1100 - 990 = 110 \]

### Step 5: Calculate Total Gain or Loss
Now, we can find the total gain or loss:
\[ \text{Total Gain or Loss} = \text{Gain from first} - \text{Loss from second} \]
\[ \text{Total Gain or Loss} = 90 - 110 = -20 \]

This indicates a loss of Rs 20.

### Step 6: Calculate the Total Cost Price
Total Cost Price:
\[ \text{Total CP} = CP_1 + CP_2 = 900 + 1100 = 2000 \]

### Step 7: Calculate Loss Percentage
Loss Percentage is calculated as:
\[ \text{Loss Percentage} = \left( \frac{\text{Total Loss}}{\text{Total CP}} \right) \times 100 \]
\[ \text{Loss Percentage} = \left( \frac{20}{2000} \right) \times 100 = 1\% \]

### Final Answer
Mr. Sinha incurred a total loss of 1%.

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