Simple & Compound Interest: Formulas, Growth, Applications

TAKEAWAYS:

• Simple interest grows linearly (fixed % of principal each year), while compound interest grows exponentially (interest on interest).
• The formula for simple interest is $SI=P\times r\times t$, and for compound interest it is $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.
• Compound interest is more common in banks, loans, and investments because it yields higher returns over time.
• The rule of 72 estimates how long it takes for an investment to double using compound interest.
• Interest rates can be nominal (stated rate) or effective (actual rate after compounding).
• Always identify whether interest is simple or compound before applying formulas.

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1. What is Interest?

Interest is the cost of borrowing money or the earnings from lending money. It is usually expressed as a percentage (rate) of the principal amount per year.

Types of Interest

There are two main types:

1. Simple Interest (SI) – Interest is calculated only on the original principal.
2. Compound Interest (CI) – Interest is calculated on the principal + previous interest.

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2. Simple Interest (SI)

Simple interest is easy to calculate and is used in short-term loans, trade discounts, and some government bonds.

Formula

$$SI=P\times r\times t$$ Where:

• $P$ = Principal (initial amount)
• $r$ = Annual interest rate (in decimal, e.g., 5% = 0.05)
• $t$ = Time in years

Amount (A) with Simple Interest

$$A=P+SI=P(1+r\times t)$$

Worked Example 1: Simple Interest Calculation

Problem: Mr. Thapa borrows NPR 50,000 at 8% per annum simple interest for 3 years. How much interest will he pay?

Solution: Given:

• $P=50,000$
• $r=8\%=0.08$
• $t=3$ years

$$SI=50,000\times 0.08\times 3=12,000$$ Answer: Mr. Thapa will pay NPR 12,000 in interest.

Worked Example 2: Finding Principal or Rate

Problem: A sum of money becomes NPR 7,500 in 2.5 years at 6% per annum simple interest. Find the principal.

Solution: Given:

• $A=7,500$
• $r=6\%=0.06$
• $t=2.5$ years

$$\begin{gathered} A=P(1+r\times t) \\ 7,500=P(1+0.06\times 2.5) \\ 7,500=P(1+0.15) \\ 7,500=1.15P \\ P=\frac{7,500}{1.15}=6,521.74\approx 6,522 \end{gathered}$$ Answer: The principal was NPR 6,522.

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3. Compound Interest (CI)

Compound interest is more common in banks, mutual funds, and long-term investments because it grows faster than simple interest.

Formula (Compounded Annually)

$$A=P{\left(1+\frac{r}{n}\right)}^{nt}$$ Where:

• $A$ = Amount after time $t$
• $P$ = Principal
• $r$ = Annual interest rate (decimal)
• $n$ = Number of times interest is compounded per year (e.g., 1 for yearly, 12 for monthly)
• $t$ = Time in years

If Compounded Annually ($n=1$)

$$A=P(1+r{)}^t$$

Worked Example 3: Compound Interest (Annual Compounding)

Problem: Mrs. Karki deposits NPR 20,000 in a bank at 5% per annum compound interest. How much will she have after 4 years?

Solution: Given:

• $P=20,000$
• $r=5\%=0.05$
• $t=4$ years
• $n=1$ (compounded yearly)

$$\begin{gathered} A=20,000{\left(1+0.05\right)}^4 \\ A=20,000\times (1.05{)}^4 \\ A=20,000\times 1.21550625 \\ A=24,310.13 \end{gathered}$$ Answer: Mrs. Karki will have NPR 24,310.13 after 4 years.

Worked Example 4: Compound Interest (Monthly Compounding)

Problem: A bank offers 6% per annum compounded monthly. If you deposit NPR 10,000, how much will it grow to in 3 years?

Solution: Given:

• $P=10,000$
• $r=6\%=0.06$
• $t=3$ years
• $n=12$ (compounded monthly)

$$\begin{gathered} A=10,000{\left(1+\frac{0.06}{12}\right)}^{12\times 3} \\ A=10,000{\left(1+0.005\right)}^{36} \\ A=10,000\times (1.005{)}^{36} \\ A=10,000\times 1.1972 \\ A=11,972 \end{gathered}$$ Answer: The amount will be NPR 11,972.

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4. Simple vs. Compound Interest: Key Differences

Observation:

• After 5 years, simple interest gives NPR 12,500.
• Compound interest gives NPR 12,763 (higher by NPR 263).

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5. The Rule of 72 (Estimating Doubling Time)

The Rule of 72 helps estimate how long it takes for an investment to double at a given interest rate.

Formula: $$\text{Years to double}=\frac{72}{\text{Interest Rate (in \%)}}$$

Worked Example 5: Using the Rule of 72

Problem: At what rate will an investment double in 8 years?

Solution: $$\text{Rate}=\frac{72}{8}=9\%$$ Answer: The investment must earn 9% per annum to double in 8 years.

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6. Nominal vs. Effective Interest Rate

• Nominal Rate = Stated interest rate (e.g., 6% per annum).
• Effective Rate = Actual rate after compounding.

Formula for Effective Rate (Annual Compounding)

$$\text{Effective Rate}=(1+r{)}^n-1$$ Where:

• $r$ = nominal rate per compounding period
• $n$ = number of compounding periods per year

Worked Example 6: Effective Interest Rate

Problem: A bank offers 12% per annum compounded quarterly. What is the effective annual rate (EAR)?

Solution: Given:

• Nominal rate = 12% = 0.12
• Compounded quarterly ($n=4$)

$$\begin{gathered} \text{EAR}={\left(1+\frac{0.12}{4}\right)}^4-1 \\ \text{EAR}=(1+0.03{)}^4-1 \\ \text{EAR}=(1.03{)}^4-1 \\ \text{EAR}=1.1255-1=0.1255\text{ or }12.55\% \end{gathered}$$ Answer: The effective annual rate is 12.55%.

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7. Applications in Business

1. Loans & Borrowings

   • Banks use compound interest for home loans, car loans.
   • Simple interest is used in short-term trade credit.

2. Investments

   • Fixed deposits, mutual funds, stocks grow via compound interest.
   • Longer investment periods lead to higher returns.

3. Business Decisions

   • Comparing loan options (which has lower effective rate?).
   • Deciding between simple discount vs. compound discount.

4. Inflation & Purchasing Power

   • If interest rate > inflation rate, real returns are positive.

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8. Common Mistakes to Avoid

❌ Mixing up simple and compound formulas. ❌ Forgetting to convert percentage to decimal (e.g., 5% = 0.05). ❌ Ignoring compounding frequency (monthly vs. yearly). ❌ Assuming all interest is compounded annually (some loans compound monthly/quarterly). ❌ Not checking if interest is added to principal (affects compound interest).

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9. Exam Tip: How to Score Full Marks

✅ Always write the formula first before plugging in numbers. ✅ Show all steps (NEB examiners reward working, not just answers). ✅ Label units clearly (NPR, years, %). ✅ Round to 2 decimal places for money (e.g., NPR 12,345.67). ✅ Compare simple vs. compound when asked for differences. ✅ Use the Rule of 72 for quick estimates in word problems.

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10. NEB-Style Practice Questions

Section A: Short Answer (1-2 marks each)

1. Define simple interest and give its formula.
2. What is the difference between nominal and effective interest rate?
3. If a sum doubles in 6 years at compound interest, what is the rate? (Use Rule of 72.)
4. Why does compound interest grow faster than simple interest?

Section B: Calculation (3-5 marks each)

5. Simple Interest Problem: A man borrows NPR 80,000 at 7.5% per annum simple interest for 4 years. Calculate:

   • (a) Total interest paid.
   • (b) Total amount to be repaid.

6. Compound Interest Problem: A bank offers 8% per annum compounded half-yearly. If you deposit NPR 50,000, how much will it be worth after 5 years?

7. Comparison Problem: Compare the amounts after 3 years for:

   • (a) NPR 10,000 at 6% simple interest.
   • (b) NPR 10,000 at 6% compound interest (annual). Which gives more? By how much?

Section C: Word Problem (5-7 marks)

8. Loan Repayment Plan: Mr. Gurung takes a loan of NPR 200,000 at 10% per annum compound interest, repayable in 3 years.
   • (a) Calculate the total amount to be repaid.
   • (b) If he repays NPR 70,000 at the end of each year, how much extra will he pay in interest compared to simple interest?

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11. Summary Table: Key Formulas

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12. Real-World Example: Bank Fixed Deposit

Scenario: You deposit NPR 50,000 in a bank at 6% per annum compounded annually for 5 years.

Calculations: $$\begin{gathered} A=50,000\times (1.06{)}^5 \\ A=50,000\times 1.3382 \\ A=66,910 \end{gathered}$$ Interest Earned: $$66,910-50,000=16,910$$ Conclusion: You earn NPR 16,910 in interest over 5 years.

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Final Notes for NEB Exam

• Memorize formulas but understand their use.
• Practice numerical problems daily.
• Check units (years vs. months, % vs. decimal).
• Draw tables to compare simple vs. compound growth.
• Use the Rule of 72 for quick answers in word problems.

Good luck! 🚀