Question

A man sold \(\frac{3}{5}\)th of his articles at a gain of 20% and the remaining at cost price. Find the percentage gain earned in the transaction.

• A. 8
• B. 10
• C. 12
• D. 14

Hint:

Use the alligation (weighted average) formula with the two gain rates and their weights.

Answer 1

The Correct Option is C

To find the overall percentage gain in the transaction, we need to analyze the gain from both parts of the sale, i.e., the portion sold at a profit and the portion sold at cost price. 

Assume the total cost price of all articles is \(C\).

The man sold \(\frac{3}{5}\)th of his articles at a gain of 20%.

Cost price of \(\frac{3}{5}\) of the articles = \(\frac{3}{5} \times C\).

Selling price of this portion = Cost price + Gain

• Gain is 20% of the cost price: \(0.2 \times \frac{3}{5} \times C = \frac{0.6}{5} \times C = \frac{6}{50} \times C\)

Thus, Selling price = Cost price + Gain = \(\frac{3}{5} \times C + \frac{6}{50} \times C = \frac{30}{50} \times C + \frac{6}{50} \times C = \frac{36}{50} \times C\)

The remaining \(\frac{2}{5}\) of the articles are sold at cost price.

• Cost price for this portion = \(\frac{2}{5} \times C\)
• Selling price = Cost price = \(\frac{2}{5} \times C\)

Calculate the total selling price:

Total selling price = Selling price of first part + Selling price of second part

= \(\frac{36}{50} \times C + \frac{2}{5} \times C\)

= \(\frac{36}{50} \times C + \frac{20}{50} \times C\)

= \(\frac{56}{50} \times C\)

Calculate the overall gain:

Overall gain = Total selling price - Total cost price

= \(\frac{56}{50} \times C - C\)

= \((\frac{56}{50} - 1) \times C = \frac{6}{50} \times C\)

Percentage gain = \(\left(\frac{Gain}{Total\ Cost\ Price} \times 100\right) \%\)

= \(\left(\frac{\frac{6}{50} \times C}{C} \times 100\right) \%\)

= \(\left(\frac{6}{50} \times 100\right) \%\)

= 12%

Thus, the percentage gain earned in the transaction is 12%. Therefore, the correct answer is 12.

Answer 2

Step 1: Note the two profit rates and their weights.
The man sells \(\frac{3}{5}\) of his articles at a 20% gain, and the remaining \(\frac{2}{5}\) at cost price, meaning 0% gain.

Step 2: Apply the weighted average (alligation) formula.
The overall gain percentage is \(p = \dfrac{p_1 q_1 + p_2 q_2}{q_1 + q_2}\), where \(p_1 = 20\), \(q_1 = \frac{3}{5}\), \(p_2 = 0\), \(q_2 = \frac{2}{5}\).

Step 3: Substitute and simplify.
\(p = \dfrac{20 \times \frac{3}{5} + 0 \times \frac{2}{5}}{\frac{3}{5} + \frac{2}{5}} = \dfrac{12}{1} = 12\).

Final Answer:
The overall percentage gain is 12%. \[ \boxed{12\%} \]