Question

If a person buys 15 pencils for ₹12, and sells 12 pencils for ₹15, find the gain or loss percentage.

• A. Loss of 56.25%
• B. Gain of 65.25%
• C. Loss of 56.52%
• D. Gain of 56.25%

Hint:

To solve this faster, use the cross-multiplication method: $(15 \times 15) - (12 \times 12) = 225 - 144 = 81$. Then Gain % = $(81 / 144) \times 100 = 56.25\%$.

Answer 1

The Correct Option is D

Step 1 : Understanding the Question
This problem falls under the category of Profit and Loss, specifically dealing with transactions where the quantity of goods bought differs from the quantity sold. To determine the financial outcome of such a trade, we must evaluate whether the revenue generated from the sale exceeds the initial investment. The primary challenge here is the discrepancy in the number of units (15 vs 12), which necessitates normalizing the quantities to a single unit or a common multiple to allow for an accurate comparison of prices.
Step 2 : Key Formulas and approach
The fundamental approach involves calculating the Cost Price (CP) and Selling Price (SP) for a single pencil (the Unitary Method). Once the unit values are established, we compare them to determine the nature of the transaction (Profit if SP>CP, or Loss if CP>SP).
Key Formulas:
1. $\text{Unit Price} = \frac{\text{Total Price}}{\text{Total Quantity}}$
2. $\text{Gain} = \text{Selling Price} - \text{Cost Price}$
3. $\text{Gain %} = \left( \frac{\text{Total Gain}}{\text{Cost Price}} \right) \times 100$
Step 3 : Detailed Explanation

Determination of Cost Price (CP): The problem states that 15 pencils were purchased for Rs12. To find the cost of one pencil, we divide the total cost by the quantity: $CP = \frac{12}{15} = Rs0.80$ per pencil.

Determination of Selling Price (SP): The person sells 12 pencils for Rs15. To find the selling price of one pencil, we divide the total sales revenue by the quantity sold: $SP = \frac{15}{12} = Rs1.25$ per pencil.

Comparative Analysis: By comparing the two values, we see that Rs1.25 is greater than Rs0.80, which confirms a profit (gain). The absolute gain per pencil is calculated as: $1.25 - 0.80 = Rs0.45$.

Calculation of Gain Percentage: To express this gain as a percentage of the initial investment, we use the profit percentage formula: $\text{Gain %} = \left( \frac{0.45}{0.80} \right) \times 100$.

Mathematical Simplification: Simplifying the fraction $\frac{0.45}{0.80}$ gives $\frac{45}{80}$, which reduces to $\frac{9}{16}$. Multiplying $\frac{9}{16}$ by 100 results in $0.5625 \times 100 = 56.25%$.


Step 4 : Final Answer
The transaction results in a total gain of 56.25%, which corresponds to option (D).