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.