Determining the Quantity for Rice Mixture

This question asks us to find out how much quantity of a cheaper rice needs to be mixed with a certain amount of dearer rice. The goal is to achieve a specific selling price for the mixture, resulting in a particular profit percentage.

Information Analysis:

• Cheaper Rice: Cost = Rs. 42 per kg. Let the required quantity be \(x\) kg.
• Dearer Rice: Cost = Rs. 50 per kg. Quantity = \(7\frac{1}{2}\) kg = 7.5 kg.
• Mixture Selling Price (SP): Rs. 53.10 per kg.
• Profit Percentage: 18%.

Step 1: Calculate the Cost Price (CP) of the Mixture

Before we can use the alligation method, we must find the actual cost price per kg of the mixture. We know the selling price and the profit percentage.

The relationship between Selling Price (SP), Cost Price (CP), and Profit Percentage is given by the formula:

SP = CP \times (1 + \frac{Profit\%}{100})

We have SP = Rs. 53.10 and Profit% = 18%. Plugging these into the formula:

53.10 = CP \times (1 + \frac{18}{100})

53.10 = CP \times (1 + 0.18)

53.10 = CP \times 1.18

To find the CP, we rearrange the equation:

CP = \frac{53.10}{1.18}

CP = 45

So, the cost price of the mixture is Rs. 45 per kg.

Step 2: Use Alligation to Find the Ratio of Quantities

Alligation is a method used to solve problems where different quantities of two ingredients are mixed to produce a mixture of a certain average quantity or price.

We set up the costs of the individual rice types and the mean cost price of the mixture:

Now, we find the difference between the costs:

• Difference between the Dearer Rice cost and Mean CP: \(50 - 45 = 5\)
• Difference between the Mean CP and Cheaper Rice cost: \(45 - 42 = 3\)

The rule of alligation states that the ratio of the quantities of the ingredients is inversely proportional to these differences.

\frac{Quantity \ of \ Cheaper \ Rice}{Quantity \ of \ Dearer \ Rice} = \frac{Difference \ for \ Dearer \ Rice}{Difference \ for \ Cheaper \ Rice}

Let the quantity of the cheaper rice be \(x\) kg. We know the quantity of the dearer rice is 7.5 kg.

\frac{x}{7.5} = \frac{5}{3}

Step 3: Calculate the Quantity of Cheaper Rice

To find \(x\), we solve the equation derived from the alligation:

x = 7.5 \times \frac{5}{3}

First, divide 7.5 by 3:

\frac{7.5}{3} = 2.5

Now, multiply the result by 5:

x = 2.5 \times 5

x = 12.5

So, 12.5 kg of rice costing Rs. 42 per kg must be mixed.

Step 4: Relate the Result to the Options

The calculated quantity is 12.5 kg. We need to express this in the format given in the options.

12.5 \text{ kg} = 12 \text{ kg} + 0.5 \text{ kg} = 12 \text{ kg} + \frac{1}{2} \text{ kg} = 12\frac{1}{2} \text{ kg}

Comparing this with the given options:

• Option 1: \(12\frac{1}{2}\) - This matches our calculated quantity.
• Option 2: \(10\frac{1}{2}\)
• Option 3: 8
• Option 4: 9
• Option 5: (Not specified)

The required quantity is \(12\frac{1}{2}\) kg.