One quality of wheat at ₹ 9.30 per kg is mixed with another quality at a certain rate in the ratio 8 : 7. If the mixture so formed be worth ₹ 10 per kg, what is the rate per kg of the second quality of wheat ?

### Concept & Weighted Average The mean value of a mixture is the weighted average of the prices of its components based on their mixing ratio. $$ \text{Mean Price} = \frac{n_1p_1 + n_2p_2}{n_1 + n_2} $$ ### Step-by-Step Solution 1. Define the given variables: - $p_1$ (price of first quality) = ₹ $9.30$ - $n_1$ (ratio proportion of first quality) = $8$ - $n_2$ (ratio proportion of second quality) = $7$ - Mean price = ₹ $10$ 2. Let the price of the second quality be $x$. 3. Set up the weighted average equation: - $\frac{(8 \times 9.30) + (7 \times x)}{8 + 7} = 10$ 4. Solve for $x$: - $\frac{74.40 + 7x}{15} = 10$ - $74.40 + 7x = 150$ - $7x = 150 - 74.40$ - $7x = 75.60$ - $x = \frac{75.60}{7} = 10.80$ ### Exam Strategy & Shortcut Use deviation from the mean: The mean is $10$. The first type is $9.30$, which is $-0.70$ below the mean. For 8 parts, total deficit $= 8 \times -0.70 = -5.60$. This deficit must be balanced by the second type. So, the 7 parts of the second type must have a total surplus of $+5.60$. Surplus per part $= \frac{5.60}{7} = +0.80$. Therefore, the price of the second type $= 10 + 0.80 = \text{₹ } 10.80$. ### Common Pitfall Miscalculating the multiplication of decimals ($8 \times 9.30$) leading to an incorrect constant in the algebraic equation. ### Final Answer Therefore, the correct answer is **₹ 10.80**.