Compound Interest Formula: Definition, Examples, and Practice Questions

Compound Interest Formula: Understanding compound interest is essential for mastering mathematics, providing a critical foundation for school curricula and competitive exams such as SSC, Railway, and Banking. This article explores the fundamentals of compound interest, including its formula, derivation, variations for different compounding periods, practical shortcuts, and detailed solved problems to test your knowledge.

Basics of Compound Interest

Compound interest is defined as the interest calculated on the initial principal and the accumulated interest from previous periods. It is a fundamental financial concept applied in everyday life. Unlike simple interest, which is calculated only on the principal, compound interest accounts for the principal plus interest earned over time, making it a powerful tool for wealth growth.

You have likely noticed that interest is regularly credited to your bank account. Because the interest is added back to the principal, the interest amount grows progressively each year. This is the hallmark of compound interest (CI). By reading further, you will understand exactly why the long-term returns from compound interest significantly outperform simple interest.

Compound Interest Formula

Compound interest is a method where interest is calculated based on both the original principal amount and the interest accumulated from earlier periods. In academic mathematics, this is standardly represented by the acronym C.I.

Compound interest serves as the backbone of modern finance and banking sectors. Some key applications of compound interest include the following points:

1. Population data varies whether increasing or decreasing.
2. The Bacterial Growth.
3. Inflation and Depreciation in the value of commodities.

Compound Interest Formula Tricks & Tips 

To calculate compound interest effectively, you must identify variables such as the Principal amount, Rate of interest, and Time period. Below are some simple tricks and formulas to streamline your calculations.

1. If a principal amount P is invested at an annual interest rate R compounded annually, the total amount A after t years is expressed as:

2. When the interest is compounded on a half-yearly (semi-annual) basis, the formula adjusts to:

3. When the interest is compounded on a quarterly basis, the formula becomes:

4. If the annual interest rates vary for consecutive years (e.g., R₁, R₂, and R₃), the final amount is calculated as:

5. If the time period is given in a mixed fraction (e.g., 2¾ years), use the fractional power formula:

6. (a) The difference between compound interest and simple interest on a fixed principal for two years at a rate R% is calculated as:

(b) The difference between compound interest and simple interest for three years at a rate R% is calculated as:

7. If a sum becomes n times its original value in t years at compound interest, it will become nᵐ times in m × t years.

8. When a sum becomes n times in t years, the effective rate of compound interest is expressed as: 

9. When a sum grows to amount x in time 'A' and to amount y in time 'B', the annual compound interest rate is calculated as:

10. For a loan of amount P at an interest rate R% repaid in 'n' equal annual installments, the installment value is calculated as:

Compound Interest Solved Questions

Question 1: A village had 50,000 residents in 2010. Its population decreases at a rate of 10% per year. What will be the total population in 2015?

Solution: The population declines at a steady rate of 10% per annum. Since each year's population is calculated based on the previous year's total, we apply the compound depreciation formula. 

For decreasing values, the formula is: A = P(1 – R/100)ⁿ

Population after 5 years (2010 to 2015) = 50000(1 – 10/100)⁵

= 50000 × (0.9)⁵ = 50000 × 0.59049 = 29524.5

Question 2: A bird population increases at a rate of 3% per day. Calculate the population after 2 days, starting from an initial count of 300,000.

Solution: As the population increases at 3% daily, we use: A = P(1 + R/100)ⁿ

Population after 2 days = 300000 (1 + 3/100)²

= 300000 × (1.03)² = 300000 × 1.0609 = 318270

Question 3: A TV costs Rs. 2500 and depreciates by 9% per month. Find its value after 5 months.

Solution: To find the depreciated value, use: A = P(1 – R/100)ⁿ

Value after 5 months = 2500 (1 – 9/100)⁵

= 2500 × (0.91)⁵ = 2500 × 0.624 ≈ Rs. 1560

Question 4: Calculate the compound interest on a loan of Rs. 5000 for 1.5 years at 20% per annum, compounded half-yearly.

Solution: Since interest is compounded half-yearly:

Principal (P) = Rs. 5000

Time (T) = 1.5 years = 3 half-years

Rate (R) = 20% / 2 = 10% per half-year

Amount (A) is calculated as:

A = P(1 + R/100)ᵀ

A = 5000 × (1 + 10/100)³

= 5000 × (1.331) = Rs. 6655

CI = A – P = Rs. 6655 – Rs. 5000 = Rs. 1655

Question 5: What is the compound interest on Rs. 3000 for 4 years at 15% per annum, compounded annually?

Solution: Principal (P) = Rs. 3000, Time (T) = 4 years, Rate (R) = 15%

Amount (A) is expressed as:

A = P(1 + R/100)ᵀ

A = 3000 (1 + 15/100)⁴ 

= 3000 × (1.15)⁴ ≈ Rs. 5247

Compound Interest = A – P = 5247 – 3000 = Rs. 2247

Question 6: Sohit borrows Rs. 20,000 for 2 years at 12% interest compounded yearly. Calculate the total amount and interest to be paid.

Solution: Given:

Principal = Rs. 20000, Rate = 12%, Time = 2 years

Using the compound interest formula:

Amount (A) = P (1 + R/100)²

Substituting the values:

A = 20000 (1 + 12/100)²

= 20000 × (1.12)² = Rs. 25088

Compound Interest = A – P = 25088 – 20000 = Rs. 5088

Question 7: The difference between simple interest and compound interest on Rs. 50,000 for 2 years is Rs. 267. Calculate the rate of interest.

Solution: Using the difference formula: CI - SI = P(R/100)²

267 = 50000 × (R/100)²

267 / 50000 = (R/100)²

R ≈ 7.30%