A business person buys potatoes of two different varieties P and Q, mixes them in a certain ratio and sells them at ₹192 per kg.
The cost of the variety P is ₹800 for 5 kg.
The cost of the variety Q is ₹800 for 4 kg.
If the person gets 8% profit, what is the P : Q ratio (by weight)?
Potato Mix Ratio Calculation
This problem requires finding the ratio of two potato varieties, P and Q, by weight, given their costs, a selling price, and a profit percentage. We will use the concept of cost price, selling price, profit, and the rule of alligation.
Calculate Cost Price per kg for Varieties P and Q
First, determine the cost price per kilogram for each variety:
- Variety P: ₹800 for 5 kg. Cost per kg = $\frac{800}{5} = ₹160$ per kg.
- Variety Q: ₹800 for 4 kg. Cost per kg = $\frac{800}{4} = ₹200$ per kg.
Determine the Mean Cost Price (CP)
The business person sells the mixture at ₹192 per kg and makes an 8% profit. The mean cost price of the mixture can be calculated using the formula: $CP = \frac{SP}{1 + \text{Profit%}}$.
- Selling Price (SP) = ₹192 per kg
- Profit = 8% = 0.08
- Mean CP = $\frac{192}{1 + 0.08} = \frac{192}{1.08}$
- Calculation: $\frac{192}{1.08} = \frac{19200}{108}$. Simplifying this fraction gives $\frac{1600}{9}$.
- So, the Mean CP of the mixture is $₹\frac{1600}{9}$ per kg.
Apply Alligation Rule to Find P : Q Ratio
We now use the rule of alligation with the cost prices of variety P, variety Q, and the mean cost price of the mixture.
- Cost Price of P ($CP_P$): ₹160 per kg
- Cost Price of Q ($CP_Q$): ₹200 per kg
- Mean Cost Price (Mean CP): $₹\frac{1600}{9}$ per kg
Apply alligation:
- Difference between $CP_Q$ and Mean CP: $200 - \frac{1600}{9} = \frac{1800 - 1600}{9} = \frac{200}{9}$
- Difference between Mean CP and $CP_P$: $\frac{1600}{9} - 160 = \frac{1600 - 1440}{9} = \frac{160}{9}$
The ratio of P to Q is given by the inverse ratio of these differences:
- Ratio P : Q = (Difference Q : Mean CP) : (Mean CP : Difference P)
- Ratio P : Q = $\frac{200}{9} : \frac{160}{9}$
- Simplifying the ratio: $200 : 160 = 20 : 16 = 5 : 4$.
Therefore, the ratio P : Q (by weight) is 5:4.
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