Introduction
Problems with equal installments appear often in exams: a borrower repays a loan in equal payments (installments) at regular intervals while simple interest is charged on the outstanding amount. This pattern is important because it combines time conversion, tracking outstanding principal, and basic algebra to find installment, principal, rate or time.
Pattern: Equal Installments in SI
Pattern
Key concept: For each installment, interest is charged on the outstanding principal for the period it remained unpaid; update outstanding after each payment and set final outstanding = 0 to form a linear equation for the installment.
Core steps:
1. Convert payment timings to years (months ÷ 12).
2. For each period: Interest = (Outstanding_before × R × T)/100.
3. After payment: Outstanding_after = Outstanding_before + Interest - Installment.
4. Repeat until final payment; set final outstanding = 0 and solve.
Step-by-Step Example
Question
A man borrows ₹6,000 and agrees to repay it in three equal annual installments at the end of each year. If the rate of simple interest is 10% per annum, find the amount of each installment.
Options:
- A. ₹2000
- B. ₹2200
- C. ₹2412
- D. ₹2600
Solution
-
Step 1: Define the unknown
Let each installment = X. -
Step 2: Record given values
Outstanding = 6000, Rate = 10% p.a., Time per installment = 1 year. -
Step 3: Compute interest each year and update outstanding
-
End of Year 1:
Interest = (6000 × 10)/100 = 600
Outstanding after payment = 6000 + 600 - X = 6600 - X -
End of Year 2:
Interest = (6600 - X) × 10% = 660 - 0.1X
Outstanding = (6600 - X) + (660 - 0.1X) - X = 7260 - 2.1X -
End of Year 3:
Interest = (7260 - 2.1X) × 10% = 726 - 0.21X
Outstanding = (7260 - 2.1X) + (726 - 0.21X) - X = 7986 - 3.31X
-
End of Year 1:
-
Step 4: Apply final condition
Set final outstanding = 0 → 7986 - 3.31X = 0. -
Step 5: Solve
X = 7986 ÷ 3.31 = 2412.08. -
Final Answer:
₹2412 → Option C -
Quick Check:
Substituting X = 2412 leaves outstanding ≈ 0 after the third payment (tiny rounding difference). ✅
Quick Variations
1. Installments half-yearly or monthly → convert to years.
2. Installments at the beginning of period → adjust interest period.
3. Unknown principal/rate → form linear equation using outstanding.
4. Large installment count → use average outstanding method.
Trick to Always Use
- Convert all installment intervals to years first.
- Compute interest using (Outstanding × R × T)/100.
- Update outstanding after each payment.
- Set final outstanding to zero and solve.
Summary
Summary
- Apply SI formula correctly for each outstanding period.
- Track outstanding balance across installments.
- Final outstanding = 0 gives the required equation.
- Works for finding installment, rate, time, or principal.
Example to remember: ₹6000 repaid in 3 equal installments at 10% → each ≈ ₹2412.
Hint: Form outstanding after each payment: Outstanding_after = Outstanding_before + Interest - Installment, then set final outstanding = 0.
Common Mistakes: Using only principal ÷ number of installments (ignores interest) or forgetting to add interest before subtracting the installment.