Calculating Rice Cost Price Ratio
Let the cost price per kg of rice A be \(C_A\) and rice B be \(C_B\). The trader mixes them in a ratio of 3:5 by weight.
Assume the trader takes 3 kg of rice A and 5 kg of rice B. The total weight is \(3 + 5 = 8\) kg.
Cost Price (CP) Calculation:
- CP of 3 kg rice A = \(3 \times C_A\)
- CP of 5 kg rice B = \(5 \times C_B\)
- Total CP of the mixture = \(3C_A + 5C_B\)
Selling Price (SP) Calculation:
- Rice A is sold at a 20% gain: \(SP_A = C_A \times (1 + 20/100) = 1.2 C_A\). The SP for 3 kg is \(3 \times 1.2 C_A = 3.6 C_A\).
- Rice B is sold at a 10% loss: \(SP_B = C_B \times (1 - 10/100) = 0.9 C_B\). The SP for 5 kg is \(5 \times 0.9 C_B = 4.5 C_B\).
- Total SP of the mixture = \(3.6 C_A + 4.5 C_B\).
Overall Profit Analysis:
- The overall profit is 8%.
- Total SP = Total CP \(\times (1 + 8/100)\)
- Total SP = \((3C_A + 5C_B) \times 1.08\)
Equating SP expressions:
- \(3.6 C_A + 4.5 C_B = (3C_A + 5C_B) \times 1.08\)
- \(3.6 C_A + 4.5 C_B = 3.24 C_A + 5.4 C_B\)
- Rearrange the terms to find the ratio:
- \(3.6 C_A - 3.24 C_A = 5.4 C_B - 4.5 C_B\)
- \(0.36 C_A = 0.9 C_B\)
- \(\frac{C_A}{C_B} = \frac{0.9}{0.36} = \frac{90}{36}\)
- Simplify the fraction: \(\frac{90}{36} = \frac{10 \times 9}{4 \times 9} = \frac{10}{4} = \frac{5}{2}\).
The ratio of the cost prices per kg of rice A to rice B is 5:2.
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