Installments & Split Investments

🎯 Learning Objective

Solve equal-installment debt questions and two-part investment problems at SI.

💡 Concept

• Equal installments: each installment paid early keeps earning SI till the debt closes
• Installment paid with k years left grows to x × (1 + kR/100)
• Debt D due after n years = sum of grown installments: x(1 + (n−1)R/100) + … + x
• Two-part investment: let one part be x, the other (Total − x); write ONE interest equation
• Total interest always lies between the two individual rates — a quick sanity check

🧮 Key Formulas

Each installment grows: x(1 + kR/100), k = years remaining

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Sum of grown installments = Debt due

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Part 1 × R₁ + Part 2 × R₂ = Total interest × 100

✏️ Easy Example

Q. A debt of ₹1,050 due 2 years from now at 10% p.a. simple interest is settled in 2 equal annual installments. Find each installment.

1. Year-1 installment grows for 1 year: 1.1x
2. Year-2 installment: x
3. 2.1x = 1050 → x = 1050 ÷ 2.1

Answer: ₹500

🇮🇳 Real-Life Example

Buying a ₹12,000 phone from a local dealer on 'aasaan kisht' — each installment quietly carries interest; this maths shows the real price you pay.

📝 Exam-Level Example

Q. What annual installment will discharge a debt of ₹1,092 due in 3 years at 12% p.a. simple interest?

1. Installments grow to 1.24x, 1.12x and x
2. Total = 3.36x = 1092
3. x = 1092 ÷ 3.36

Answer: ₹325

📝 Exam-Level Example

Q. ₹10,000 is invested partly at 8% and partly at 12% simple interest. The total annual interest is ₹1,040. Find the part invested at 12%.

1. Let the 12% part be x
2. 0.12x + 0.08(10000 − x) = 1040
3. 0.04x = 240 → x = 6000

Answer: ₹6,000

🪄 Memory Trick

Installments: factors are 1, 1 + R/100, 1 + 2R/100 … add them once, divide the debt — no year-by-year drama.

⚠️ Common Mistakes

• ❌ Multiplying every installment by the full n years instead of the years remaining
• ❌ Taking the average of the two rates in split-investment questions
• ❌ Forgetting the last installment carries zero interest

🏆 Exam Tips

• ✅ For n installments at R%: total factor = n + R/100 × (0+1+…+(n−1))
• ✅ In split investments, assume x on the rate being ASKED — fewer steps at the end

📌 Summary

• Each installment grows by its remaining years
• Sum of grown installments = debt due
• Split investment → one equation in x
• Blended interest always sits between the two rates