Question

A salesman sells two kinds of trousers: cotton and woollen. A pair of cotton trousers is sold at 30% profit and a pair of woollen trousers is sold at 50% profit. The salesman had calculated that if he sells 100% more woollen trousers than cotton trousers, his overall profit would be 45%. However, he ends up selling 50% more cotton trousers than woollen trousers. What will be his overall profit?

• A. 37.5%
• B. 40%
• C. 41%
• D. 42.33%

Hint:

Overall profit is a cost-weighted average; first find the cost ratio of woollen to cotton trousers from the planned 45% profit, then reapply it with the actual quantity ratio.

Answer 1

The Correct Option is B

Step 1: Set up profit as a weighted average.
Overall profit percentage is a weighted average of the two profit rates, where the weight of each type is its share of total cost, not its share of quantity. Let the cost of a cotton trouser be $x$ and a woollen trouser be $y$.

Step 2: Use the planned scenario to connect $x$ and $y$.
Planned woollen quantity is double the cotton quantity. If cotton quantity is $a$, woollen quantity is $2a$.
\[ \frac{0.30(ax) + 0.50(2ay)}{ax + 2ay} = 0.45 \]
Cancel $a$ and simplify:
\[ 0.30x + 1.00y = 0.45x + 0.90y \]
\[ 0.10y = 0.15x \implies y = 1.5x \]
A woollen trouser's cost price is 1.5 times a cotton trouser's cost price.

Step 3: Apply this ratio to the actual sales.
Actual cotton quantity is 50% more than woollen quantity, so if woollen quantity is $b$, cotton quantity is $1.5b$. With $y = 1.5x$:
\[ \frac{0.30(1.5b \cdot x) + 0.50(b \cdot 1.5x)}{1.5b \cdot x + b \cdot 1.5x} \]

Step 4: Simplify the fraction.
Numerator: $0.45bx + 0.75bx = 1.20bx$. Denominator: $1.5bx + 1.5bx = 3.0bx$.
\[ \text{Overall profit} = \frac{1.20bx}{3.0bx} = 0.40 = 40\% \]

Final Answer:
The actual overall profit is 40%, so the correct choice is B. \[ \boxed{40\%} \]