A man lends ₹ $10,000$ in four parts. If he gets $8\%$ on ₹ $2000$; $7\frac{1}{2}\%$ on ₹ $4000$ and $8\frac{1}{2}\%$ on ₹ $1400$; what percent must he get for the remainder, if his average annual interest is $8.13\%$ ?

### Concept & Weighted Average Rate The total average interest earned is the sum of the interests earned from each individual part. This can be expressed as a linear equation where the total interest equals the sum of the partial interests. ### Step-by-Step Solution - Total Principal = ₹ $10,000$. - Average annual interest rate = $8.13\%$. - Total annual interest = $\frac{10000 \times 8.13}{100} = 813$. - Let's calculate the interest from the first three parts: 1. Interest on ₹ $2000$ at $8\%$ = $\frac{2000 \times 8}{100} = 160$. 2. Interest on ₹ $4000$ at $7.5\%$ = $\frac{4000 \times 7.5}{100} = 300$. 3. Interest on ₹ $1400$ at $8.5\%$ = $\frac{1400 \times 8.5}{100} = 119$. - Total interest from these three parts = $160 + 300 + 119 = 579$. - Find the remaining principal: Remainder = $10000 - (2000 + 4000 + 1400) = 10000 - 7400 = 2600$. - Let the required rate for the remainder be $R\%$. - Interest on the remainder = Total Interest - Interest from first 3 parts - $\frac{2600 \times R}{100} = 813 - 579$ - $26R = 234$ - $R = \frac{234}{26} = 9$ - The required rate is $9\%$. ### Exam Strategy & Shortcut You can balance the deviations from the average rate ($8.13\%$). Part 1 ($2000$): deviation is $-0.13\%$. Total = $2000 \times (-0.13) = -260$ Part 2 ($4000$): deviation is $-0.63\%$. Total = $4000 \times (-0.63) = -2520$ Part 3 ($1400$): deviation is $+0.37\%$. Total = $1400 \times (+0.37) = +518$ Sum of deviations = $-260 - 2520 + 518 = -2262$. To balance to $0$, the 4th part ($2600$) must contribute $+2262$. Required deviation for $2600$ = $\frac{2262}{2600} = +0.87\%$. Required rate = Average + Deviation = $8.13\% + 0.87\% = 9.00\%$. ### Common Pitfall Calculation errors when dealing with mixed fractions like $7\frac{1}{2}\%$ and decimals like $8.13\%$. Converting mixed fractions to pure decimals ($7.5\%$) usually makes the arithmetic cleaner. ### Final Answer Therefore, the correct answer is **9%**.