Aditya wants to discharge a debt of Rs. 5643, due in 6 years, by paying equal…

Concept

When a debt is discharged by paying T equal annual installments, each installment (except the last) accumulates simple interest until the due date. For a debt (Amount) A due after T years at R% per annum simple interest, the value of each annual installment I is found by equating the sum of all installments plus the interest each earns by the due date to A, which gives the standard formula: I = (100 × A) / [100T + R × T × (T − 1)/2].

Application

Given: Debt (Amount) A = Rs. 5643, Time T = 6 years, Rate R = 4% per annum.

1. Write the installment formula: I = (100 × A) / [100T + R × T × (T − 1)/2].

2. Substitute A = 5643, T = 6 and R = 4: I = (100 × 5643) / [100 × 6 + (4 × 6 × 5)/2].

3. Simplify the denominator: 100 × 6 = 600, and (4 × 6 × 5)/2 = 60, so the denominator is 600 + 60 = 660.

4. Simplify the numerator: 100 × 5643 = 564300.

5. Divide: I = 564300 / 660 = 855.

Cross-check

Verify independently: six installments of Rs. 855 are paid at the end of years 1 to 6, so they accumulate simple interest for 5, 4, 3, 2, 1 and 0 years respectively by year 6. Their total value at the due date is (6 × 855) + 855 × (4/100) × (5+4+3+2+1+0) = 5130 + 855 × 0.04 × 15 = 5130 + 513 = 5643, which matches the debt exactly — confirming that Rs. 855 is the correct annual installment.