Question

A merchant mixes two varieties of rice, one costing Rs.12.75 per kg with another variety costing Rs.12 per kg in the proportion 1: r. He sells the mixture at Rs. 13.5C per kg and there by gains 10%. Find the value of r.

• A. 2
• B. 1.75
• C. 2.5
• D. 2.75
• E. 1.5

Answer 1

The Correct Option is B

Step 1: Understand the problem.
A merchant mixes two varieties of rice, one costing Rs. 12.75 per kg and the other Rs. 12 per kg in the proportion 1: r. He sells the mixture at Rs. 13.50 per kg and gains 10%. We need to find the value of \( r \).

Step 2: Use the cost price and selling price relationship.
The selling price of the mixture is Rs. 13.50 per kg, and the merchant gains 10%. This means the cost price (C.P.) of the mixture is 90% of the selling price. The relationship between the selling price (S.P.) and cost price (C.P.) is given by:
\[ \text{C.P.} = \frac{\text{S.P.}}{1 + \text{Gain percent}} \] Substituting the given values:
\[ \text{C.P.} = \frac{13.50}{1 + 0.10} = \frac{13.50}{1.10} = 12.27 \, \text{per kg} \] Therefore, the cost price of the mixture is Rs. 12.27 per kg.

Step 3: Set up the equation for the cost price of the mixture.
The cost price of the mixture is the weighted average of the cost prices of the two varieties of rice. The first variety costs Rs. 12.75 per kg, and the second variety costs Rs. 12 per kg. The ratio in which they are mixed is 1: r.
The cost price of the mixture is given by the formula:
\[ \text{C.P.} = \frac{12.75 \times 1 + 12 \times r}{1 + r} \] Equating this to the cost price calculated earlier (Rs. 12.27), we get: \[ \frac{12.75 \times 1 + 12 \times r}{1 + r} = 12.27 \] Simplifying the equation: \[ \frac{12.75 + 12r}{1 + r} = 12.27 \] Multiplying both sides by \( 1 + r \): \[ 12.75 + 12r = 12.27(1 + r) \] Expanding both sides: \[ 12.75 + 12r = 12.27 + 12.27r \] Bringing like terms together: \[ 12.75 - 12.27 = 12.27r - 12r \] \[ 0.48 = 0.27r \] Solving for \( r \): \[ r = \frac{0.48}{0.27} \approx 1.75 \]

Step 4: Conclusion.
The value of \( r \) is 1.75.

Final Answer:
The correct option is (B): 1.75.