## 1. What Is Compound Interest? **Compound interest (CI)** is interest calculated on the original principal **plus the interest already added** at the end of previous compounding periods. In simple words, it is *interest on interest*.
**Simple interest (SI):** Interest is always calculated on the original principal. **Compound interest:** At the end of each compounding period, interest is added to the principal. The next period's interest is calculated on the increased amount.
Example: On ₹10,000 at 10% per year, the first year's interest is ₹1,000. With annual compounding, the second year's interest is calculated on ₹11,000, not ₹10,000.
## 2. Symbols and the Basic Formula - **P:** Principal (original money) - **R:** Annual interest rate, expressed as a percentage - **n:** Time in years - **A:** Total amount, including interest - **CI:** Compound interest
**For annual compounding:** **A = P × (1 + R/100)^n** **CI = A − P**
The rate and the number of periods must match. If an annual rate of 12% is compounded every six months, the rate for each six-month period is 6%.
## 3. Worked Example 1 – Year-by-Year Method **Question:** Find the compound interest on ₹10,000 at 10% per year for two years, compounded annually.
First-year interest = 10,000 × 10/100 = **₹1,000**. Amount after year 1 = 10,000 + 1,000 = **₹11,000**. Second-year interest = 11,000 × 10/100 = **₹1,100**. Total compound interest = 1,000 + 1,100 = **₹2,100**. Final amount = **₹12,100**.
**Comparison:** Simple interest for two years would be ₹2,000. Compound interest is ₹100 higher.
## 4. Worked Example 2 – Three Years Using the Formula **Question:** Find the compound interest on ₹8,000 at 5% per year for three years.
A = 8,000 × (1 + 5/100)^3 = 8,000 × (1.05)^3 = 8,000 × 1.157625 = **₹9,261**. CI = 9,261 − 8,000 = **₹1,261**.
**Shortcut:** First calculate the final amount A, then subtract the original principal P.
## 5. Half-Yearly, Quarterly and Monthly Compounding | Compounding frequency | Rate per period | Number of periods | Amount formula | |---|---|---|---| | Annually | R% | n | P(1 + R/100)^n | | Every six months | R/2% | 2n | P(1 + R/200)^(2n) | | Every three months | R/4% | 4n | P(1 + R/400)^(4n) | | Monthly | R/12% | 12n | P(1 + R/1200)^(12n) |
Here R is the **annual** percentage rate and n is the number of **years**. Adjust both the rate and the number of periods together.
## 6. Worked Example 3 – Half-Yearly Compounding **Question:** Find the compound interest on ₹8,000 at 10% per year for one year, compounded every six months.
Rate per six months = 10/2 = **5%**. Number of six-month periods = **2**. A = 8,000 × (1.05)^2 = 8,000 × 1.1025 = **₹8,820**. CI = 8,820 − 8,000 = **₹820**.
**Common mistake:** Do not apply the full 10% rate separately in both six-month periods.
## 7. Worked Example 4 – Quarterly Compounding **Question:** Find the compound interest on ₹16,000 at 20% per year for nine months, compounded every three months.
Rate per quarter = 20/4 = **5%**. Number of quarters = 9/3 = **3**. A = 16,000 × (1.05)^3 = 16,000 × 1.157625 = **₹18,522**. CI = 18,522 − 16,000 = **₹2,522**.
## 8. Worked Example 5 – Two and a Half Years, Annual Compounding **Question:** Calculate interest on ₹4,000 at 10% per year for 2½ years, with interest compounded **annually** and simple interest applied to the remaining half-year.
Amount after the first two complete years: A₂ = 4,000 × (1.10)^2 = **₹4,840**. Simple interest for the remaining half-year = 4,840 × 10 × ½ / 100 = **₹242**. Final amount = 4,840 + 242 = **₹5,082**. Total interest = 5,082 − 4,000 = **₹1,082**.
**Important assumption:** This is the standard school-mathematics convention for annual compounding with an incomplete final year. Follow the stated contract or compounding rule if the question specifies something different.
## 9. Worked Example 6 – Difference Between CI and SI **Question:** Find the difference between compound interest and simple interest on ₹5,000 at 8% per year for two years.
For **two years with annual compounding and a constant rate**: **CI − SI = P × (R/100)^2**. = 5,000 × (8/100)^2 = 5,000 × 0.0064 = **₹32**.
Check: SI = 5,000 × 8 × 2 / 100 = ₹800. CI = 5,000 × (1.08)^2 − 5,000 = ₹832. Difference = 832 − 800 = **₹32**.
**Do not apply this two-year shortcut unchanged to three-year problems.**
## 10. Worked Example 7 – Find the Principal **Question:** A sum becomes ₹2,028 after two years at 4% compound interest per year. Find the principal.
P = A / (1 + R/100)^n = 2,028 / (1.04)^2 = 2,028 / 1.0816 = **₹1,875**.
Check: 1,875 × 1.0816 = ₹2,028.
## 11. Worked Example 8 – Different Rates Each Year **Question:** Find the compound interest on ₹15,000 if the annual rates are 15% in year 1, 20% in year 2 and 25% in year 3.
A = 15,000 × 1.15 × 1.20 × 1.25 = **₹25,875**. CI = 25,875 − 15,000 = **₹10,875**.
**Rule:** Multiply the growth factor for each year. Do not add 15%, 20% and 25% and treat the total as a single annual rate.
## 12. Worked Example 9 – Depreciation and Population Growth **(a) Depreciation:** A machine worth ₹16,800 loses 25% of its value each year for two years. Remaining value = 16,800 × (1 − 25/100)^2 = 16,800 × 0.75² = **₹9,450**.
**(b) Reverse population growth:** A town's population increases by 10% every year and is now 26,620. What was its population three years ago? P = 26,620 / (1.10)^3 = 26,620 / 1.331 = **20,000 people**.
**Growth formula:** P(1 + R/100)^n. **Depreciation formula:** P(1 − R/100)^n.
## 13. Exam Shortcuts and Common Mistakes 1. **A = P(1 + R/100)^n** gives the **total amount**, not the interest. Calculate **CI = A − P**. 2. For half-yearly compounding, **halve the annual rate and double the number of years into periods**. 3. For quarterly compounding, **divide the annual rate by four and multiply the number of years by four**. 4. The two-year shortcut **CI − SI = P(R/100)^2** assumes annual compounding at a constant rate. 5. For incomplete years, check **how often interest is added** before choosing the method. 6. If the rate changes every year, multiply the corresponding yearly growth factors. 7. Round money to two decimal places when appropriate, preferably only after completing the main calculation.
### Simple Interest vs Compound Interest | Simple Interest (SI) | Compound Interest (CI) | |---|---| | Interest on original principal only | Interest on principal plus added interest | | I = PRT/100 | CI = P(1 + R/100)^n − P | | Yearly interest stays the same at a fixed rate | Interest generally increases each compounding period | | ₹10,000 at 10% for 2 years → ₹2,000 | ₹10,000 at 10% for 2 years → ₹2,100 |
## 14. One-Minute Revision **Annual compounding:** A = P(1 + R/100)^n; CI = A − P. **Half-yearly:** A = P(1 + R/200)^(2n). **Quarterly:** A = P(1 + R/400)^(4n). **Two-year CI−SI difference:** P(R/100)^2. **Depreciation:** P(1 − R/100)^n. **Find principal:** P = A/(1 + R/100)^n.
## 15. Textbook References and Complete Book-Back Tracking **Primary textbook reference:** Tamil Nadu Samacheer Kalvi, **Class 8 Mathematics, Term 2, Unit 1 – Life Mathematics, Exercise 1.3**. Topics include annual, half-yearly and quarterly compound interest, growth, depreciation and CI–SI differences.
A publicly available textbook guide indexes **20 numbered questions**. Question 1 has five subparts and Question 2 has five subparts, suggesting **28 separately answerable items in that guide**. However, **the complete original 2026–27 government textbook exercise has not been directly verified**. The guide also contains some transcription and mathematical errors; its solutions must not be treated as official answer keys.
**Complete original exercise:** Source not yet verified. **Completed and verified against the original textbook:** **0**. **Pending:** Full original count not yet confirmed. **Planned batches:** Once the original textbook is checked, questions will be scheduled in numbered evening batches, including every subquestion.
All nine worked examples in today's lesson are **original or adapted instructional examples**, not labelled as verified TNPSC previous-year questions or original book-back items. No TNPSC PYQ is claimed because an official paper/year/question reference and final-key match have not been verified for this lesson.
## 16. Revision Plan and Next Session **Revise today's Compound Interest lesson:** 20 October (+1 day), 22 October (+3), 26 October (+7) and 2 November (+14). **Revise today:** Day 06 Simple Interest (+1 day) and Day 04 Ratio and Proportion (+3 days). **7:00 PM practice:** Exactly **15 headline MCQs** on today's Tamil grammar (8), Motion (5) and Compound Interest (2), with book-back and weekly review in separate additional sections.
### ETN English Push Notification **Title:** 🔢 TNPSC Group IV | Compound Interest Made Easy! **Message:** 📚 Learn compound interest, SI vs CI, half-yearly and quarterly formulas, and nine fully worked examples. Complete Day 07 Aptitude notes on ETN News!
