Calculating the Ratio of Mixed Rice Quantities

This problem involves mixing two types of rice with different costs to achieve a specific average cost for the mixture. We are given the price per kg for Quality A rice (Rs 35), Quality B rice (Rs 65), and the desired average price of the mixture (Rs 50).

Understanding Mixture Problems and Alligation

Mixture problems often deal with combining two or more items with different properties (like price, concentration, etc.) to get a mixture with a desired average property. The rule of alligation is a helpful technique to solve such problems, especially when finding the ratio of the quantities of the components.

The rule of alligation states that if two ingredients are mixed, the ratio of their quantities is inversely proportional to the difference between their individual price/concentration and the mean price/concentration of the mixture.

Applying the Rule of Alligation to Rice Mixture

Let's apply the rule of alligation to find the ratio of the quantities of Quality A and Quality B rice in the mixture. We place the prices of the two types of rice on the left and right sides and the average price of the mixture in the middle.

According to the rule of alligation, the ratio of the quantity of Rice A to the quantity of Rice B will be the ratio of the differences we calculated, but in a cross-wise manner. Specifically, the quantity of Rice A is proportional to the difference $|65 - 50|$, and the quantity of Rice B is proportional to the difference $|50 - 35|$.

Ratio of Quantity A : Quantity B = (Difference B) : (Difference A)

Ratio of Quantity A : Quantity B = $|65 - 50| : |50 - 35|$

Ratio of Quantity A : Quantity B = $15 : 15$

Simplifying the Quantity Ratio

The ratio $15 : 15$ can be simplified by dividing both parts by their greatest common divisor, which is 15.

Ratio of Quantity A : Quantity B = $\frac{15}{15} : \frac{15}{15} = 1 : 1$

Thus, the ratio of the quantity of Quality A rice to the quantity of Quality B rice in the mixture is 1 : 1.

Verification using Weighted Average

Let $Q_A$ be the quantity of rice A and $Q_B$ be the quantity of rice B. The total cost of the mixture is $35 Q_A + 65 Q_B$. The total quantity is $Q_A + Q_B$. The average price is given by:

$$ \text{Average Price} = \frac{\text{Total Cost}}{\text{Total Quantity}} $$

$$ 50 = \frac{35 Q_A + 65 Q_B}{Q_A + Q_B} $$

Multiplying both sides by $(Q_A + Q_B)$:

$$ 50(Q_A + Q_B) = 35 Q_A + 65 Q_B $$

$$ 50 Q_A + 50 Q_B = 35 Q_A + 65 Q_B $$

Rearranging the terms to group $Q_A$ and $Q_B$:

$$ 50 Q_A - 35 Q_A = 65 Q_B - 50 Q_B $$

$$ 15 Q_A = 15 Q_B $$

Dividing both sides by $15 Q_B$ (assuming $Q_B \neq 0$):

$$ \frac{15 Q_A}{15 Q_B} = \frac{15 Q_B}{15 Q_B} $$

$$ \frac{Q_A}{Q_B} = \frac{1}{1} $$

So, the ratio $Q_A : Q_B$ is $1 : 1$. This confirms the result obtained using the rule of alligation.

Conclusion on Rice Mixture Ratio

The ratio of the quantity of Quality A rice to the quantity of Quality B rice required to achieve an average price of Rs 50 per kg is 1 : 1.

Revision Table: Key Concepts for Mixture Problems

Additional Information: Weighted Averages in Mixtures

The average price of a mixture is a type of weighted average. If you mix a quantity $Q_1$ of an item with price $P_1$ and a quantity $Q_2$ of an item with price $P_2$, the average price of the mixture ($P_{avg}$) is calculated as:

$$ P_{avg} = \frac{(P_1 \times Q_1) + (P_2 \times Q_2)}{Q_1 + Q_2} $$

In our rice example, $P_1 = 35$, $P_2 = 65$, and $P_{avg} = 50$. We were looking for the ratio $Q_1 : Q_2$. The algebraic method shown earlier demonstrates how this weighted average formula is used to derive the ratio.

Weighted averages are commonly used in various fields, such as calculating average grades (where different assignments have different weights), average speed over different distances, or average density of a composite material.