A shopkeeper sells two T.V. sets at the same price. There is a gain of 20% on one TV and a loss of 20% on the other. State which of the following statement is correct :

Options

A. The shopkeeper makes no net gain or profit
B. The shopkeeper loses by 2%
C. The shopkeeper gains by 4 %
D. The shopkeeper loses by 4%

Answer

To solve the problem step by step, we will analyze the situation of the shopkeeper selling two T.V. sets at the same price, one with a gain of 20% and the other with a loss of 20%.

### Step 1: Understand the Selling Price
Let’s assume the selling price of each T.V. set is \( SP \).

### Step 2: Calculate the Cost Price of Each T.V.
1. For the first T.V. set, which has a gain of 20%:
- Let the cost price (CP) of the first T.V. be \( CP_1 \).
- The selling price can be expressed as:
\[ SP = CP_1 + 0.20 \times CP_1 = 1.20 \times CP_1 \]
- Rearranging gives:
\[ CP_1 = \frac{SP}{1.20} \]

2. For the second T.V. set, which has a loss of 20%:
- Let the cost price of the second T.V. be \( CP_2 \).
- The selling price can be expressed as:
\[ SP = CP_2 - 0.20 \times CP_2 = 0.80 \times CP_2 \]
- Rearranging gives:
\[ CP_2 = \frac{SP}{0.80} \]

### Step 3: Calculate the Total Cost Price
Now, we need to find the total cost price of both T.V.s:
\[ Total\ CP = CP_1 + CP_2 = \frac{SP}{1.20} + \frac{SP}{0.80} \]

### Step 4: Find a Common Denominator
To add these fractions, we find a common denominator:
- The least common multiple of 1.20 and 0.80 is 2.40.
- Convert each fraction:
\[ CP_1 = \frac{SP}{1.20} = \frac{SP \times 2}{2.40} = \frac{2SP}{2.40} \]
\[ CP_2 = \frac{SP}{0.80} = \frac{SP \times 3}{2.40} = \frac{3SP}{2.40} \]

### Step 5: Add the Cost Prices
Now we can add:
\[ Total\ CP = \frac{2SP}{2.40} + \frac{3SP}{2.40} = \frac{(2SP + 3SP)}{2.40} = \frac{5SP}{2.40} \]

### Step 6: Calculate the Overall Gain or Loss
Now, we need to compare the total selling price with the total cost price:
- Total Selling Price for both T.V.s:
\[ Total\ SP = SP + SP = 2SP \]

- Total Cost Price:
\[ Total\ CP = \frac{5SP}{2.40} \]

### Step 7: Determine Gain or Loss Percentage
To find out if there is a gain or loss, we can calculate:
\[ Loss = Total\ CP - Total\ SP = \frac{5SP}{2.40} - 2SP \]
Convert \( 2SP \) to have the same denominator:
\[ 2SP = \frac{4.80SP}{2.40} \]
Thus,
\[ Loss = \frac{5SP}{2.40} - \frac{4.80SP}{2.40} = \frac{0.20SP}{2.40} \]

### Step 8: Calculate Loss Percentage
Now, the loss percentage can be calculated as:
\[ Loss\ Percentage = \left( \frac{Loss}{Total\ CP} \right) \times 100 \]
Substituting the values:
\[ Loss\ Percentage = \left( \frac{0.20SP}{\frac{5SP}{2.40}} \right) \times 100 = \left( \frac{0.20 \times 2.40}{5} \right) \times 100 = \frac{0.48}{5} \times 100 = 9.6\% \]

However, we can also use the quick formula mentioned in the video:
\[ Loss\ Percentage = \frac{Gain \times Loss}{100} = \frac{20 \times 20}{100} = 4\% \]

### Conclusion
The shopkeeper incurs an overall loss of 4%.