Problem Statement: A company borrows ` 10,000 on condition to repay it with compound interest at 5% p.a by annual installments of ` 1000 each. The number of years by which the debt will be clear is?

Solution:

Step 1: Calculate the interest on the principal amount

- Formula: A = P(1 + r/n)^(nt)
- where A = final amount, P = principal amount, r = annual interest rate, n = number of times the interest is compounded per year, t = time in years
- In this case, P = ` 10,000, r = 5%, n = 1 (compounded annually), t = time in years
- Therefore, A = ` 10,000(1 + 0.05/1)^(1t)
- Simplifying, A = ` 10,500

Step 2: Calculate the remaining amount after first installment

- The company pays ` 1000 as the first installment, leaving a balance of ` 9,500
- This amount will now earn interest for the remaining time period

Step 3: Calculate the time required to pay off the remaining amount

- Formula: A = P(1 + r/n)^(nt)
- In this case, P = ` 9,500, r = 5%, n = 1 (compounded annually), A = ` 0 (since the remaining amount needs to be paid off completely), t = time in years
- Therefore, ` 0 = ` 9,500(1 + 0.05/1)^(1t)
- Simplifying, 1.05^t = 1.0
- Taking logarithm on both sides, t = log(1.0)/log(1.05)
- Therefore, t = 14.21 years (approx.)

Step 4: Add the time required for the first installment

- The first installment was paid at the beginning of the first year, so the total time required to pay off the debt is 14.21 + 1 = 15.21 years (approx.)

Answer: The debt will be clear in approximately 15.21 years.