A shopkeeper mixes two varieties of rice, R 1 and R 2, in the ratio 7 : 5.
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A shopkeeper mixes two varieties of rice, $R_1$ and $R_2$, in the ratio $7 : 5$. The cost price of variety $R_2$ is Rs. 22 per kg. If the cost price of the blended mixture is Rs. 18 per kg, what is the cost price per kg of variety $R_1$?
Show answer & explanation
The weighted average of the cost prices of the two rice varieties must equal the cost price of the mixture. Using the weighted average formula, we solve for the unknown cost price of variety R1 based on the given ratio and the final mixture price.
Step-by-step Derivation:
Step 1: Define variables. Let the cost price of variety $R_1$ be $x$ Rs/kg. The cost price of variety $R_2$ is given as 22 Rs/kg. The ratio of $R_1$ to $R_2$ is $7:5$.
Step 2: Set up the weighted average equation. The cost price of the mixture is the total cost divided by the total weight: $\frac{(7 \times x) + (5 \times 22)}{7 + 5} = 18$.
Step 3: Simplify the equation: $\frac{7x + 110}{12} = 18$.
Step 4: Multiply both sides by 12: $7x + 110 = 18 \times 12 = 216$.
Step 5: Isolate $7x$: $7x = 216 - 110 = 106$.
Step 6: Solve for $x$: $x = \frac{106}{7} \approx 15.1428$.
Step 7: Rounding to two decimal places, we get Rs. 15.14.