Question

A dealer sold two types of goods for 10,000 each. On one of them, he lost 20% and on the other he gained 20%. His gain or loss percent in the entire transaction was ?

• A. 2% loss
• B. 2% gain
• C. 4% gain
• D. 4% loss

Hint:

When two items are sold at the same price, and one has a P% profit and the other a P% loss, there is *always* a net loss. The loss percentage is simply $\left(\frac{P}{10}\right)^2\%$. Remember this shortcut for efficiency in competitive exams.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem describes a transaction where a dealer sells two items at the same selling price, but incurs a loss on one and a profit on the other, both at the same percentage rate. We need to calculate the overall gain or loss percentage for the entire transaction.

Step 2: Key Formula or Approach:
There are two ways to approach this:
1. Detailed Calculation: Calculate the Cost Price (CP) for each item, then find the total CP and compare it with the total Selling Price (SP).
2. Shortcut Formula: When two items are sold at the same Selling Price, and one is sold at an x% profit and the other at an x% loss, there is always a net loss. The loss percentage is given by the formula:
\[ \text{Loss \%} = \left(\frac{x}{10}\right)^2 \% \]
Where x is the common profit/loss percentage.

Step 3: Detailed Explanation:
Given:
- Selling Price (SP) of each good = Rs. 10,000.
- Percentage Loss on one good = 20%.
- Percentage Gain on the other good = 20%.
Method 1: Detailed Calculation
1. Total Selling Price:
Total SP = SP1 + SP2 = Rs. 10,000 + Rs. 10,000 = Rs. 20,000.
2. Cost Price of the first good (with 20% loss):
Let CP1 be the cost price of the first good.
SP1 = CP1 - (20% of CP1) = CP1 (1 - 0.20) = 0.80 \(\times\) CP1
Rs. 10,000 = 0.80 \(\times\) CP1
CP1 = Rs. $\frac{10,000}{0.80}$ = Rs. 12,500.
3. Cost Price of the second good (with 20% gain):
Let CP2 be the cost price of the second good.
SP2 = CP2 + (20% of CP2) = CP2 (1 + 0.20) = 1.20 \(\times\) CP2
Rs. 10,000 = 1.20 \(\times\) CP2
CP2 = Rs. $\frac{10,000}{1.20}$ = Rs. $\frac{100,000}{12}$ = Rs. $\frac{25,000}{3}$ \(\approx\) Rs. 8,333.33.
4. Total Cost Price:
Total CP = CP1 + CP2 = Rs. $12,500 + \frac{25,000}{3}$ = Rs. $\frac{37,500 + 25,000}{3}$ = Rs. $\frac{62,500}{3}$ \(\approx\) Rs. 20,833.33.
5. Overall Gain or Loss:
Since Total SP (Rs. 20,000) \textless Total CP (Rs. 20,833.33), there is an overall loss.
Loss amount = Total CP - Total SP = Rs. $\frac{62,500}{3} - 20,000$ = Rs. $\frac{62,500 - 60,000}{3}$ = Rs. $\frac{2,500}{3}$.
6. Loss Percentage:
Loss % = $\left(\frac{\text{Loss amount}}{\text{Total CP}}\right) \times 100$
Loss % = $\left(\frac{2,500/3}{62,500/3}\right) \times 100$
Loss % = $\left(\frac{2,500}{62,500}\right) \times 100$
Loss % = $\left(\frac{25}{625}\right) \times 100$ = $\left(\frac{1}{25}\right) \times 100$ = 4%.
Method 2: Using the Shortcut Formula
Given x = 20%.
\[ \text{Loss \%} = \left(\frac{x}{10}\right)^2 \% \]
\[ \text{Loss \%} = \left(\frac{20}{10}\right)^2 \% = (2)^2 \% = 4\% \]
Since this formula always results in a loss under these conditions, the result is a 4% loss.

Step 4: Final Answer:
The dealer incurred a 4% loss in the entire transaction.

Answer 2

Instead of plugging in the specific selling price of Rs. 10,000 right away, this method sets up the problem with general symbols first and only substitutes numbers at the very end, which also makes it easy to check every option.

1. 2% loss: Let the common selling price of each item be \( S \) and the common profit/loss percentage be \( x \). The cost price of the item sold at a gain is \( \text{CP}_1 = \frac{S}{1+\frac{x}{100}} \) and the cost price of the item sold at a loss is \( \text{CP}_2 = \frac{S}{1-\frac{x}{100}} \). Adding these, \( \text{Total CP} = S\left(\frac{1}{1+\frac{x}{100}}+\frac{1}{1-\frac{x}{100}}\right) = \frac{2S}{1-\left(\frac{x}{100}\right)^2} \), which is always greater than \( 2S \) (the total selling price) whenever \( x \neq 0 \). This means the transaction always ends in a loss, never a 2% or any gain, so a "loss" figure smaller than what the algebra gives is not correct either; this option undershoots the true loss.
2. 2% gain: As shown above, whenever equal percentages are used, one for profit and one for loss, on two items with the same selling price, the outcome is always a net loss, never a gain, no matter what the common percentage is. So a gain of any size is impossible here.
3. 4% gain: For the same reason, a gain cannot occur in this setup, so this option is also ruled out.
4. 4% loss: Using the formula derived above, the overall loss percentage works out to \( \text{Loss \%} = \frac{\text{Total CP}-\text{Total SP}}{\text{Total CP}}\times 100 = \left(\frac{x}{100}\right)^2 \times 100 = \frac{x^2}{100} \). Substituting \( x = 20 \), \( \text{Loss \%} = \frac{20^2}{100} = \frac{400}{100} = 4\% \). This matches exactly.

The algebra confirms that whenever equal-percentage profit and equal-percentage loss are applied to two items sold at the same price, the transaction always results in a net loss equal to the square of the common percentage divided by 100, which here gives 4%.

Therefore, the correct answer is 4% loss.