Understanding the Rice Mixture Problem

This question involves mixing two different qualities of rice, each with a specific price per kilogram (kg), in a given ratio. The goal is to find the price per kg of the second quality of rice, given the price of the first quality, the mixing ratio, and the price of the resulting mixture.

This type of problem can be solved using the concept of weighted average or alligation. The principle is that the total value of the quantities being mixed equals the value of the total quantity of the mixture.

Setting Up the Calculation

Let's define the variables:

• Price of the first quality of rice = Rs. 45 per kg.
• Ratio of the first quality in the mixture = 3 parts.
• Price of the second quality of rice = \(x\) per kg (this is what we need to find).
• Ratio of the second quality in the mixture = 2 parts.
• Price of the mixture = Rs. 50 per kg.

The total parts in the mixture are the sum of the individual ratios, which is \(3 + 2 = 5\) parts.

Calculating the Rate of the Second Quality Rice

We can set up an equation based on the weighted average of the prices. The total value contributed by the first quality is its price multiplied by its ratio (or quantity proportional to the ratio). Similarly for the second quality. The sum of these values must equal the total value of the mixture (mixture price multiplied by the total ratio or quantity).

Mathematically, this can be written as:

\[ (\text{Price}_1 \times \text{Ratio}_1) + (\text{Price}_2 \times \text{Ratio}_2) = (\text{Mixture Price} \times \text{Total Ratio}) \]

Substitute the known values into the equation:

\[ (45 \times 3) + (x \times 2) = (50 \times (3 + 2)) \]

Simplify the equation:

\[ 135 + 2x = 50 \times 5 \]

\[ 135 + 2x = 250 \]

Now, we need to solve for \(x\):

Subtract 135 from both sides of the equation:

\[ 2x = 250 - 135 \]

\[ 2x = 115 \]

Divide by 2 to find the value of \(x\):

\[ x = \frac{115}{2} \]

\[ x = 57.5 \]

So, the rate per kg of the second quality of rice is Rs. 57.5.

Let's quickly verify the result using the given options. The calculated value, Rs. 57.5, is one of the options.

Step-by-Step Solution Summary

1. Identify the prices and ratios of the two qualities of rice and the mixture price.
2. Set up the equation using the weighted average formula: \((\text{Price}_1 \times \text{Ratio}_1) + (\text{Price}_2 \times \text{Ratio}_2) = (\text{Mixture Price} \times (\text{Ratio}_1 + \text{Ratio}_2))\).
3. Substitute the given values: \((45 \times 3) + (x \times 2) = (50 \times (3 + 2))\).
4. Simplify and solve the equation for \(x\): \(135 + 2x = 250 \implies 2x = 115 \implies x = 57.5\).
5. State the final answer for the rate of the second quality rice.

Revision Table: Rice Mixture Concepts

Additional Information: Alligation Method Explained

The alligation method provides an alternative way to visualize and solve such problems. It is based on the differences in prices.

• Write the price of the cheaper quality (45) and the dearer quality (\(x\)) on the left and right sides, respectively.
• Write the mixture price (50) in the center.
• Find the difference between the mixture price and the cheaper price: \(50 - 45 = 5\). Write this difference opposite the dearer price (\(x\)). This difference represents the ratio of the dearer quality (Ratio 2).
• Find the difference between the dearer price (\(x\)) and the mixture price (50): \(x - 50\). Write this difference opposite the cheaper price (45). This difference represents the ratio of the cheaper quality (Ratio 1).

The ratios obtained this way are inversely proportional to the price differences from the mean (mixture price).

So, the ratio of cheaper quality (Ratio 1) : ratio of dearer quality (Ratio 2) is given by:

\[ (x - 50) : (50 - 45) \]

We are given the ratio is 3 : 2.

\[ (x - 50) : 5 = 3 : 2 \]

This gives us the proportion:

\[ \frac{x - 50}{5} = \frac{3}{2} \]

Now, solve for \(x\):

\[ 2 \times (x - 50) = 3 \times 5 \]

\[ 2x - 100 = 15 \]

\[ 2x = 15 + 100 \]

\[ 2x = 115 \]

\[ x = \frac{115}{2} \]

\[ x = 57.5 \]

Both the weighted average method and the alligation method yield the same result, confirming that the rate per kg of the second quality of rice is Rs. 57.5.