Simple and Compound Interest — Study Notes

Overview

Simple and Compound Interest is a high-scoring topic in SOF IMO that connects everyday banking and finance with mathematical computation. Students must master the formulas for SI and CI, understand the difference between them, and solve multi-step word problems involving principal, rate, time, and amount. This topic frequently appears in the Everyday Mathematics section and occasionally in the Achievers Section when combined with other concepts like profit-loss or time-work.

The key challenge is not just applying formulas but interpreting real-world scenarios: loans, deposits, recurring payments, and instalment schemes. Problems may involve finding any unknown variable (principal, rate, time, or amount), comparing SI and CI on the same sum, or breaking down payments into instalments. Strong arithmetic and algebraic manipulation skills are essential.

Expect 2–4 questions directly on this topic in the exam, with marks ranging from straightforward one-step calculations to multi-layered HOTS problems worth 3–4 marks each.

Key Concepts

• **Simple Interest (SI)** is calculated only on the original principal for the entire period. The interest earned each year remains constant.

• **Compound Interest (CI)** is calculated on the principal plus accumulated interest from previous periods. Money "grows on itself," leading to exponential growth.

• **Amount (A)** is the total sum returned at the end of the period: Amount = Principal + Interest.

• **Rate (R)** is always expressed as a percentage per annum unless stated otherwise (half-yearly, quarterly compounding changes the effective rate).

• **Time (T)** is measured in years unless specified otherwise. For half-yearly compounding, double the time and halve the rate; for quarterly, quadruple the time and quarter the rate.

• **Instalment problems** require working backward: the present value of each instalment (discounted back to the start) must sum to the principal borrowed.

• **Difference between CI and SI** over the same period on the same principal at the same rate increases with time and can be used to find unknowns.

• When interest is compounded annually but time is fractional (e.g., 2.5 years), compound for whole years and apply simple interest for the remaining fraction.

Formulas / Key Facts

• **Simple Interest**: SI = (P × R × T) / 100

• **Amount under SI**: A = P + SI = P(1 + RT/100)

• **Compound Interest (annual compounding)**: A = P(1 + R/100)^T, then CI = A – P

• **Half-yearly compounding**: A = P(1 + R/200)^(2T)

• **Quarterly compounding**: A = P(1 + R/400)^(4T)

• **CI for fractional time (n whole years + fraction f)**: A = P(1 + R/100)^n × (1 + fR/100)

• **Difference between CI and SI for 2 years**: CI – SI = P(R/100)²

• **Difference between CI and SI for 3 years**: CI – SI = P(R/100)²(3 + R/100)

• **Instalment formula**: Each instalment paid after time t has present value = Instalment / (1 + R/100)^t

Worked Examples

**Example 1: Basic SI calculation** A sum of ₹8000 is invested at 5% per annum simple interest for 3 years. Find the interest and amount.

*Solution*: P = 8000, R = 5%, T = 3 years SI = (8000 × 5 × 3) / 100 = 120000 / 100 = ₹1200 Amount = P + SI = 8000 + 1200 = ₹9200

**Example 2: Finding principal using CI** What sum will amount to ₹6615 in 2 years at 5% per annum compound interest?

*Solution*: A = 6615, R = 5%, T = 2 years A = P(1 + R/100)^T 6615 = P(1 + 5/100)² 6615 = P(1.05)² 6615 = P × 1.1025 P = 6615 / 1.1025 = ₹6000

**Example 3: Instalment problem** A man borrows ₹10000 at 10% per annum compound interest and agrees to repay in two equal annual instalments. Find each instalment.

*Solution*: Let each instalment be ₹x, paid at end of year 1 and year 2. Present value of 1st instalment = x / (1.1) Present value of 2nd instalment = x / (1.1)² Sum of present values = Principal x/1.1 + x/1.21 = 10000 1.21x + x = 10000 × 1.1 × 1.21 2.21x = 13310 x = 13310 / 2.21 ≈ ₹6021 (approximately)

More precise: x/1.1 + x/1.21 = 10000 → x(1/1.1 + 1/1.21) = 10000 → x(1.1 + 1)/1.21 = 10000 → x × 2.1/1.21 = 10000 → x = (10000 × 1.21)/2.1 ≈ ₹5761.90 (check calculation per exam standard)

Common Mistakes

• **Confusing SI and CI formulas** → Always check if interest is simple or compound. If not mentioned, assume compound for real-world banking scenarios; simple for straightforward loan problems.

• **Forgetting to subtract principal to find CI** → CI = A – P. Students often report the amount as the interest itself.

• **Incorrect handling of half-yearly/quarterly compounding** → Remember: double the periods and halve the rate for half-yearly; quadruple periods and quarter the rate for quarterly. Don't just plug in original R and T.

• **Using wrong time units** → If time is in months, convert to years: T(years) = months/12. Mixing units breaks the formula.

• **Rounding too early in multi-step problems** → Keep at least two decimal places during intermediate steps. Round only the final answer to avoid cumulative errors, especially in instalment problems.

Quick Reference

• SI grows linearly; CI grows exponentially — CI always exceeds SI for T > 1.
• For 2-year difference: CI – SI = P(R/100)² — useful for finding P or R quickly.
• Half-yearly compounding: double time, halve rate → A = P(1 + R/200)^(2T).
• Instalment present value = Instalment / (1 + R/100)^time — sum these to equal principal.
• If rate or time not given, use CI – SI difference formulas or algebraic setup.
• Always write down P, R, T, and which formula to use before substituting numbers.