In what ratio must a grocer mix sugar at 72 paise per kg with sugar at 48 paise per kg. So that by selling the mixture at 63 paise per kg he may gain 1/6 of his outlay ?

Options

A. 1 : 3
B. 3 : 1
C. 2 : 3
D. 3 : 2

Answer

To solve the problem of mixing two types of sugar at different prices, we can use the method of alligation. Here’s a step-by-step solution:

### Step 1: Determine the Selling Price and Gain
The selling price of the mixture is given as 63 paise per kg. The grocer wants to gain \( \frac{1}{6} \) of his outlay (cost price).

### Step 2: Calculate the Cost Price of the Mixture
To find the cost price (CP) of the mixture, we need to consider the gain. If the selling price (SP) is 63 paise and the gain is \( \frac{1}{6} \) of the cost price, we can set up the equation:

\[ SP = CP + \text{Gain} \]

The gain can be expressed as:

\[ \text{Gain} = \frac{1}{6} \times CP \]

Substituting this into the SP equation gives:

\[ 63 = CP + \frac{1}{6} CP \]

This can be simplified to:

\[ 63 = \frac{7}{6} CP \]

Now, solving for CP:

\[ CP = 63 \times \frac{6}{7} = 54 \text{ paise per kg} \]

### Step 3: Set Up the Alligation
Now we have the cost prices of the two types of sugar:
- Sugar A: 72 paise per kg
- Sugar B: 48 paise per kg
- Mean Cost (CP of mixture): 54 paise per kg

Using the alligation method, we set up the following:

\[ \text{Difference between Sugar A and Mean Cost} = 72 - 54 = 18 \]
\[ \text{Difference between Sugar B and Mean Cost} = 54 - 48 = 6 \]

### Step 4: Calculate the Ratio
The ratio of the quantities of Sugar A and Sugar B can be determined by the differences calculated above:

\[ \text{Ratio} = \frac{18}{6} = 3:1 \]

### Conclusion
The grocer must mix the sugar at 72 paise per kg with the sugar at 48 paise per kg in the ratio of 3:1.

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