Simple and Compound Interest — Study Notes

Overview

Interest problems appear in every UP Police Constable exam and typically contribute 2–4 questions. This is a scoring topic because the formulas are straightforward and questions follow predictable patterns. Mastering Simple Interest (SI) and Compound Interest (CI) — especially compounding annually and half-yearly — is essential. Additionally, instalment-based questions test whether you can apply interest formulas in multi-step scenarios. Unlike topics requiring lengthy calculations, interest problems can be solved quickly if you know the core formulas and can identify which one applies. Focus on recognising the problem type, selecting the right formula, and executing calculations accurately within exam time constraints.

The key distinction: SI remains constant each year because interest is calculated only on the principal, while CI grows because interest itself earns interest in subsequent periods. Questions may ask you to find principal, rate, time, or the difference between CI and SI. Instalment problems require you to reverse-calculate the present value of future payments. This topic rewards methodical practice and formula recall.

Key Concepts

• **Simple Interest (SI)** — Interest calculated only on the original principal throughout the loan/investment period. The interest amount stays the same every year.
• **Compound Interest (CI)** — Interest calculated on principal plus accumulated interest from previous periods. The interest amount increases each period.
• **Principal (P)** — The initial amount borrowed or invested.
• **Rate (R)** — Annual interest rate expressed as a percentage per annum.
• **Time (T)** — Duration of the loan or investment, typically in years. For half-yearly compounding, adjust both rate and time.
• **Amount (A)** — Total money at the end, which equals Principal + Interest. For SI: A = P + SI. For CI: A is calculated directly using the CI formula.
• **Compounding frequency** — Annually means interest is added once per year. Half-yearly means twice per year (divide rate by 2, multiply time by 2). Quarterly means four times per year (divide rate by 4, multiply time by 4).
• **Difference between CI and SI** — For the same principal, rate and time, CI is always greater than SI (except for 1 year, when they're equal). The difference grows with time and is a common exam question.

Formulas / Key Facts

**Simple Interest:**

• SI = (P × R × T) / 100
• Amount = P + SI = P(1 + RT/100)
• P = (SI × 100) / (R × T)
• R = (SI × 100) / (P × T)
• T = (SI × 100) / (P × R)

**Compound Interest (Annual Compounding):**

• A = P(1 + R/100)^T
• CI = A – P = P[(1 + R/100)^T – 1]

**Compound Interest (Half-Yearly Compounding):**

• A = P(1 + R/200)^(2T)
• CI = P[(1 + R/200)^(2T) – 1]
• Here R is divided by 2 and T is multiplied by 2 because interest is applied twice per year.

**Difference between CI and SI for 2 years:**

• CI – SI = P(R/100)²
• This formula saves time when only the difference is asked, not CI or SI individually.

**Difference between CI and SI for 3 years:**

• CI – SI = P(R/100)²(3 + R/100)

**Instalments (Equal Annual Payments):**

• If instalments I are paid at year-end for n years at rate R%, then: P = I / (1 + R/100) + I / (1 + R/100)² + ... + I / (1 + R/100)^n
• Each instalment is discounted back to present value.

Worked Examples

**Example 1: Simple Interest** A sum of ₹8000 is invested at 5% per annum SI for 3 years. Find the interest earned and the total amount.

*Solution:*

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

**Example 2: Compound Interest (Annual)** Find the compound interest on ₹10,000 at 10% per annum for 2 years compounded annually.

*Solution:*

• P = 10000, R = 10%, T = 2 years
• A = P(1 + R/100)^T = 10000(1 + 10/100)² = 10000(1.1)² = 10000 × 1.21 = 12100
• CI = A – P = 12100 – 10000 = ₹2100

*Alternatively, using difference formula:*

• SI for 2 years = (10000 × 10 × 2)/100 = 2000
• CI – SI = P(R/100)² = 10000(10/100)² = 10000 × 0.01 = 100
• CI = SI + 100 = 2000 + 100 = ₹2100

**Example 3: Half-Yearly Compounding** Calculate CI on ₹5000 at 8% per annum for 1 year compounded half-yearly.

*Solution:*

• P = 5000, R = 8%, T = 1 year
• For half-yearly: A = P(1 + R/200)^(2T) = 5000(1 + 8/200)² = 5000(1.04)² = 5000 × 1.0816 = 5408
• CI = 5408 – 5000 = ₹408

*Note:* If compounded annually, CI would be 5000 × 0.08 = ₹400. Half-yearly compounding gives slightly more.

**Example 4: Instalment Problem** A person borrows ₹10,000 at 10% per annum CI and repays in 2 equal annual instalments. Find each instalment.

*Solution:*

• Let each instalment be I.
• Present value of instalments = P
• 10000 = I/(1.1) + I/(1.1)²
• 10000 = I/1.1 + I/1.21
• 10000 = I(1/1.1 + 1/1.21) = I(0.9091 + 0.8264) = I(1.7355)
• I = 10000/1.7355 ≈ ₹5762 (approximately)

Common Mistakes

• **Confusing SI and CI formulas** — Students often use SI formula for CI problems or vice versa. Always check whether the problem mentions "compound" or just "interest" (usually means SI unless stated otherwise). → Read the question carefully; "compound" is the keyword.

• **Forgetting to adjust for half-yearly compounding** — Many students use the annual CI formula even when compounding is half-yearly or quarterly. → Remember: half-yearly means R/2 and 2T; quarterly means R/4 and 4T.

• **Using T in months or days without conversion** — If time is given in months or days, convert to years (months/12 or days/365) before applying the formulas. → Always standardise T to years unless otherwise specified.

• **Calculating only CI when amount is asked, or vice versa** — Questions asking for "amount" need A = P + CI, not just CI. → Identify whether the question asks for interest, amount, or both.

• **Arithmetic errors in power calculations** — Computing (1.1)³ or similar powers incorrectly is common under time pressure. → Practice squaring and cubing decimals; use approximations when exact values aren't needed.

• **Ignoring the CI–SI difference shortcut for 2 years** — Students calculate both CI and SI separately and subtract, wasting time. → Use the formula CI – SI = P(R/100)² directly for 2-year problems.

Quick Reference

• **SI = PRT/100; Amount = P + SI**
• **CI (annual): A = P(1 + R/100)^T; CI = A – P**
• **CI (half-yearly): Use R/2 and 2T in the formula**
• **CI – SI (2 years) = P(R/100)²; saves calculation time**
• **For instalments: discount each payment back to present value using (1 + R/100)^n**
• **CI > SI for periods longer than 1 year; they're equal at T = 1 year**