Understanding the Wheat Mixture Problem

This question involves a classic mixture problem where three varieties of wheat with different prices are mixed in a specific ratio. We are given the prices of two varieties, the ratio in which all three are mixed, and the final price of the resulting mixture. Our goal is to find the price of the third variety of wheat.

Let's break down the given information:

• Price of Variety 1 Wheat: Rs. 80 per kg
• Price of Variety 2 Wheat: Rs. 50 per kg
• Ratio of mixing (Variety 1 : Variety 2 : Variety 3): 1 : 2 : 3
• Price of the final Mixture: Rs. 75 per kg

We need to find the price of Variety 3 Wheat.

Setting Up the Calculation for Mixture Price

When different quantities of items with different costs are mixed, the total cost of the mixture is the sum of the costs of the individual items. The price of the mixture is the total cost divided by the total quantity.

Let the quantities of the three varieties of wheat mixed be in the ratio 1 : 2 : 3. We can represent these quantities as $\text{1k}$, $\text{2k}$, and $\text{3k}$ kilograms respectively, where $\text{k}$ is a common positive multiplier.

• Quantity of Variety 1: $\text{1k}$ kg
• Quantity of Variety 2: $\text{2k}$ kg
• Quantity of Variety 3: $\text{3k}$ kg

The total quantity of the mixture is the sum of these quantities:

Total Quantity = $\text{1k} + \text{2k} + \text{3k} = \text{6k}$ kg

Now, let's calculate the total cost of the mixture. We multiply the quantity of each variety by its respective price.

• Cost of Variety 1: $(\text{1k} \text{ kg}) \times (\text{Rs. 80/kg}) = \text{80k}$ Rupees
• Cost of Variety 2: $(\text{2k} \text{ kg}) \times (\text{Rs. 50/kg}) = \text{100k}$ Rupees
• Let the price of Variety 3 be $\text{P3}$ Rupees per kg.
• Cost of Variety 3: $(\text{3k} \text{ kg}) \times (\text{P3} \text{ Rs./kg}) = \text{3k} \times \text{P3}$ Rupees

The total cost of the mixture is the sum of these costs:

Total Cost = $\text{80k} + \text{100k} + \text{3kP3} = \text{180k} + \text{3kP3}$ Rupees

The price of the mixture is given as Rs. 75 per kg. We can set up an equation using the formula: Mixture Price = Total Cost / Total Quantity.

$\text{75} = \frac{\text{Total Cost}}{\text{Total Quantity}}$

$\text{75} = \frac{\text{180k} + \text{3kP3}}{\text{6k}}$

Solving for the Price of the Third Variety

Now, we solve the equation for $\text{P3}$, the price of the third variety.

$\text{75} = \frac{\text{k}(180 + \text{3P3})}{\text{6k}}$

Assuming $\text{k} \neq 0$ (since quantities are involved), we can cancel $\text{k}$ from the numerator and denominator:

$\text{75} = \frac{180 + \text{3P3}}{6}$

Multiply both sides of the equation by 6:

$\text{75} \times 6 = 180 + \text{3P3}$

$\text{450} = 180 + \text{3P3}$

Subtract 180 from both sides:

$\text{450} - 180 = \text{3P3}$

$\text{270} = \text{3P3}$

Divide both sides by 3:

$\text{P3} = \frac{270}{3}$

$\text{P3} = 90$

So, the price of the third variety of wheat is Rs. 90 per kg.

Verification

Let's verify if this price yields a mixture worth Rs. 75 per kg when mixed in the ratio 1:2:3. Assume quantities are 1 kg, 2 kg, and 3 kg.

• Cost of 1 kg of Variety 1: Rs. 80
• Cost of 2 kg of Variety 2: $2 \times 50 =$ Rs. 100
• Cost of 3 kg of Variety 3: $3 \times 90 =$ Rs. 270

Total Cost = $80 + 100 + 270 =$ Rs. 450

Total Quantity = $1 + 2 + 3 =$ 6 kg

Mixture Price = $\frac{\text{Total Cost}}{\text{Total Quantity}} = \frac{450}{6} =$ Rs. 75 per kg

This matches the given mixture price, confirming our calculation is correct.

Revision Table: Key Concepts

Additional Information: Weighted Average and Alligation

This type of problem is fundamentally about calculating a weighted average. The price of the mixture is the weighted average of the prices of the individual varieties, where the weights are the quantities (or the ratio parts representing quantities).

In this case, the mixture price $\text{P}_{\text{mixture}}$ is given by:

$\text{P}_{\text{mixture}} = \frac{(\text{Ratio}_1 \times \text{Price}_1) + (\text{Ratio}_2 \times \text{Price}_2) + (\text{Ratio}_3 \times \text{Price}_3)}{\text{Ratio}_1 + \text{Ratio}_2 + \text{Ratio}_3}$

Plugging in the values:

$\text{75} = \frac{(1 \times 80) + (2 \times 50) + (3 \times \text{P3})}{1 + 2 + 3}$

$\text{75} = \frac{80 + 100 + \text{3P3}}{6}$

$\text{75} = \frac{180 + \text{3P3}}{6}$

This is the same equation we solved earlier, leading to $\text{P3} = 90$.

While the Alligation rule is often used for mixtures of two components, the principle extends to more components through the weighted average concept. For three components, it's usually easier to directly apply the weighted average formula as shown above.

The final answer is Rs. 90.