For Question 9, the principal donation is P = ₹ 5000, the annual interest rate is R = 11%, and time T = 1 year. Using the simple interest formula SI = (P × R × T)/(100), the interest earned each year is (5000 × 11 × 1)/(100) = ₹ 550. This annual interest is equally distributed among 10 scholarships, giving (550)/(10) = ₹ 55 for each scholarship.

For Question 9, the principal donation is P = ₹ 5000, the annual interest rate is R = 11%, and time T = 1 year. Using the simple interest formula SI = (P × R × T)/(100), the interest earned each year is (5000 × 11 × 1)/(100) = ₹ 550. This annual interest is equally distributed among 10 scholarships, giving (550)/(10) = ₹ 55 for each scholarship.

For Question 10, the principal is P = ₹ 150. To double itself, the amount becomes A = ₹ 300, which means the simple interest required is SI = A - P = ₹ 150. Using the formula for time T = (SI × 100)/(P × R) with rate R = 4%, we compute T = (150 × 100)/(150 × 4) = 25 years.

For Question 10, the principal is P = ₹ 150. To double itself, the amount becomes A = ₹ 300, which means the simple interest required is SI = A - P = ₹ 150. Using the formula for time T = (SI × 100)/(P × R) with rate R = 4%, we compute T = (150 × 100)/(150 × 4) = 25 years.

For Question 11, let the principal sum of money be P. Since the sum doubles in T = 20 years, the simple interest earned equals the principal, so SI = P. Applying the rate formula R = (SI × 100)/(P × T), we get R = (100)/(20) = 5% per annum.

For Question 11, let the principal sum of money be P. Since the sum doubles in T = 20 years, the simple interest earned equals the principal, so SI = P. Applying the rate formula R = (SI × 100)/(P × T), we get R = (100)/(20) = 5% per annum.

For Question 12, the principal is P = ₹ 250, the amount is A = ₹ 285, and the time period is T = 2 years. The simple interest earned over 2 years is SI = 285 - 250 = ₹ 35. Using the rate formula R = (SI × 100)/(P × T), we calculate R = (35 × 100)/(250 × 2) = (3500)/(500) = 7% per annum.

For Question 12, the principal is P = ₹ 250, the amount is A = ₹ 285, and the time period is T = 2 years. The simple interest earned over 2 years is SI = 285 - 250 = ₹ 35. Using the rate formula R = (SI × 100)/(P × T), we calculate R = (35 × 100)/(250 × 2) = (3500)/(500) = 7% per annum.

9. Randhir donates ₹ $5000$ to a school, the interest of which is to be used for awarding $10$ scholarships of equal value every year. If the donation earns an interest of $11\%$ per annum, find the value of each scholarship. 10. In how many years will ₹ $150$ double itself at $4\%$ simple interest? 11. At what rate of simple interest will a sum of money double itself in $20$ years? 12. If ₹ $250$ amounts to ₹ $285$ in $2$ years, find the rate per cent per annum.

Answer: 9. ₹ 55 per scholarship; 10. 25 years; 11. 5% per annum; 12. 7% per annum

Step-by-step solution

Step 1: Solve Question 9: Value of each scholarship

For Question 9, the principal donation is $P = \text{₹} 5000$ , the annual interest rate is $R = 11\%$ , and time $T = 1\text{ year}$ . Using the simple interest formula $\text{SI} = \frac{P \times R \times T}{100}$ , the interest earned each year is $\frac{5000 \times 11 \times 1}{100} = \text{₹} 550$ . This annual interest is equally distributed among $10$ scholarships, giving $\frac{550}{10} = \text{₹} 55$ for each scholarship.

Step 2: Solve Question 10: Time required to double

For Question 10, the principal is $P = \text{₹} 150$ . To double itself, the amount becomes $A = \text{₹} 300$ , which means the simple interest required is $\text{SI} = A - P = \text{₹} 150$ . Using the formula for time $T = \frac{\text{SI} \times 100}{P \times R}$ with rate $R = 4\%$ , we compute $T = \frac{150 \times 100}{150 \times 4} = 25\text{ years}$ .

Step 3: Solve Question 11: Rate of interest to double in 20 years

For Question 11, let the principal sum of money be $P$ . Since the sum doubles in $T = 20\text{ years}$ , the simple interest earned equals the principal, so $\text{SI} = P$ . Applying the rate formula $R = \frac{\text{SI} \times 100}{P \times T}$ , we get $R = \frac{100}{20} = 5\%$ per annum.

Step 4: Solve Question 12: Rate per cent per annum

For Question 12, the principal is $P = \text{₹} 250$ , the amount is $A = \text{₹} 285$ , and the time period is $T = 2\text{ years}$ . The simple interest earned over $2$ years is $\text{SI} = 285 - 250 = \text{₹} 35$ . Using the rate formula $R = \frac{\text{SI} \times 100}{P \times T}$ , we calculate $R = \frac{35 \times 100}{250 \times 2} = \frac{3500}{500} = 7\%$ per annum.

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