Compound Interest

Class 8Comparing Quantities

Compound interest is one of the most important financial concepts taught in Class 8 mathematics. Unlike simple interest, where the interest is calculated only on the original principal amount, compound interest is calculated on the principal plus the interest that has already been accumulated. This means that with compound interest, you earn "interest on interest," causing your money to grow faster over time. Banks, savings accounts, loans, and investments all use compound interest. Albert Einstein reportedly called compound interest the "eighth wonder of the world" because of its powerful effect on wealth growth. In this chapter, you will learn what compound interest is, how it differs from simple interest, the formula for calculating it, and how to solve a variety of problems. Understanding compound interest will not only help you in your Class 8 exams but will also give you practical knowledge for managing money in real life.

What is Compound Interest?

Compound Interest (CI) is the interest calculated on the initial principal and also on the accumulated interest from previous periods. In other words, at the end of each compounding period (usually one year), the interest earned is added to the principal, and the next period's interest is calculated on this new, larger principal. This process of earning interest on interest is called "compounding."

Key Terms:

Principal (P): The initial amount of money that is borrowed, lent, or invested. This is the starting amount before any interest is applied.

Rate of Interest (R): The percentage at which interest is charged or earned per year (per annum). For example, 10% per annum means Rs 10 of interest for every Rs 100 of principal per year.

Time (n): The number of years (or compounding periods) for which the money is borrowed or invested.

Amount (A): The total money at the end of the given time period, including the original principal and all accumulated interest. The relationship is: A = P + CI.

Compound Interest (CI): The total interest earned over the entire time period. It is the difference between the final amount and the original principal: CI = A - P.

Compounding Period: The interval at which interest is calculated and added to the principal. Common periods are annual (once a year), half-yearly (twice a year), and quarterly (four times a year). More frequent compounding leads to higher total interest.

Key difference from Simple Interest: In simple interest (SI), the interest is always calculated on the original principal throughout the entire time period — the base never changes. In compound interest, the interest earned in each period is added to the principal, so the base for interest calculation grows every period. This is why compound interest always yields more than simple interest when the time is more than one period at the same rate. For exactly one year (compounded annually), CI and SI are equal.

Compound Interest Formula

The formulas for compound interest are:

1. Amount (Compounded Annually):
A = P x (1 + R/100)^n
where P = principal, R = rate of interest per annum (%), n = number of years.

2. Compound Interest:
CI = A - P = P x [(1 + R/100)^n - 1]

3. Amount (Compounded Half-Yearly):
A = P x (1 + R/200)^(2n)
Here, the rate is halved and the number of periods is doubled.

4. Amount (Compounded Quarterly):
A = P x (1 + R/400)^(4n)
Here, the rate is divided by 4 and the number of periods is multiplied by 4.

5. Finding CI when amounts for consecutive years are known:
CI for the nth year = A at end of nth year - A at end of (n-1)th year.

6. Difference between CI and SI for 2 years:
CI - SI = P x (R/100)^2
This formula helps compare CI and SI directly.

7. When rates are different for different years:
A = P x (1 + R1/100) x (1 + R2/100) x (1 + R3/100)...
where R1, R2, R3 are the rates for years 1, 2, 3, etc.

Derivation and Proof

Let us derive the compound interest formula step by step.

Given: Principal = P, Rate = R% per annum, Time = n years, compounded annually.

Year 1:
Interest for Year 1 = P x R/100
Amount at the end of Year 1 = P + P x R/100 = P(1 + R/100)

Year 2:
The principal for Year 2 is the amount at the end of Year 1 = P(1 + R/100).
Interest for Year 2 = P(1 + R/100) x R/100
Amount at the end of Year 2 = P(1 + R/100) + P(1 + R/100) x R/100
= P(1 + R/100)(1 + R/100)
= P(1 + R/100)^2

Year 3:
The principal for Year 3 = P(1 + R/100)^2.
Interest for Year 3 = P(1 + R/100)^2 x R/100
Amount at end of Year 3 = P(1 + R/100)^2 + P(1 + R/100)^2 x R/100
= P(1 + R/100)^2 x (1 + R/100)
= P(1 + R/100)^3

Generalising for n years:
By observing the pattern, after n years:
A = P(1 + R/100)^n

And Compound Interest = A - P = P[(1 + R/100)^n - 1].

Why CI exceeds SI after the first year:
In Year 1, both CI and SI give the same interest (P x R/100). But from Year 2 onwards, CI calculates interest on the accumulated amount (which is larger than P), while SI still calculates interest on the original P. This difference grows with each year, making CI progressively larger than SI.

Derivation of CI - SI for 2 years:
SI for 2 years = 2 x P x R/100 = 2PR/100
CI for 2 years = P(1 + R/100)^2 - P = P[(1 + R/100)^2 - 1]
= P[1 + 2R/100 + R^2/10000 - 1]
= P[2R/100 + R^2/10000]
= 2PR/100 + PR^2/10000
CI - SI = PR^2/10000 = P(R/100)^2

Types and Properties

Compound interest problems can be classified into several types:

1. Finding the Amount and CI (basic):
Given P, R, and n, find A and CI using the formula. This is the most common type.

2. Finding the Principal:
Given A (or CI), R, and n, find P. Rearrange the formula: P = A / (1 + R/100)^n.

3. Finding the Rate:
Given P, A, and n, find R. Use: (1 + R/100)^n = A/P, then solve for R.

4. Finding the Time:
Given P, A, and R, find n. This often involves trial or logarithms (at higher levels).

5. Comparing CI and SI:
Find the difference between compound interest and simple interest for the same P, R, and n.

6. Half-yearly and quarterly compounding:
When interest is compounded more frequently than annually, adjust the rate and time accordingly.

7. Different rates for different years:
When the interest rate changes each year, multiply the growth factors for each year.

8. Applications:
Population growth, depreciation of value, and growth of bacteria — all use the compound interest formula with modifications.

Solved Examples

Example 1: Example 1: Basic CI calculation

Problem: Find the compound interest on Rs 10,000 at 8% per annum for 2 years, compounded annually.

Solution:
P = 10,000, R = 8%, n = 2
A = P(1 + R/100)^n
A = 10,000(1 + 8/100)^2
A = 10,000(1.08)^2
A = 10,000 x 1.1664
A = Rs 11,664

CI = A - P = 11,664 - 10,000 = Rs 1,664.

Answer: The compound interest is Rs 1,664.

Example 2: Example 2: Year-by-year calculation

Problem: Find the CI on Rs 5,000 at 10% per annum for 3 years by calculating year by year.

Solution:
Year 1: Interest = 5,000 x 10/100 = 500. Amount = 5,500.
Year 2: Interest = 5,500 x 10/100 = 550. Amount = 6,050.
Year 3: Interest = 6,050 x 10/100 = 605. Amount = 6,655.

CI = 6,655 - 5,000 = Rs 1,655.

Verification using formula: A = 5,000(1.1)^3 = 5,000 x 1.331 = 6,655. CI = 1,655. Matches.

Answer: The compound interest for 3 years is Rs 1,655.

Example 3: Example 3: Comparing CI and SI

Problem: Find the difference between CI and SI on Rs 8,000 at 5% per annum for 2 years.

Solution:
Simple Interest: SI = P x R x T / 100 = 8,000 x 5 x 2 / 100 = Rs 800.

Compound Interest:
A = 8,000(1 + 5/100)^2 = 8,000(1.05)^2 = 8,000 x 1.1025 = Rs 8,820.
CI = 8,820 - 8,000 = Rs 820.

Difference: CI - SI = 820 - 800 = Rs 20.

Using the shortcut formula: CI - SI = P(R/100)^2 = 8,000 x (5/100)^2 = 8,000 x 0.0025 = Rs 20. Matches.

Answer: The difference between CI and SI is Rs 20.

Example 4: Example 4: Finding the principal

Problem: A sum of money amounts to Rs 13,230 in 2 years at 5% per annum compound interest. Find the principal.

Solution:
A = P(1 + R/100)^n
13,230 = P(1 + 5/100)^2
13,230 = P(1.05)^2
13,230 = P x 1.1025
P = 13,230 / 1.1025
P = Rs 12,000

Answer: The principal is Rs 12,000.

Example 5: Example 5: Different rates for different years

Problem: Find the amount on Rs 20,000 for 2 years if the rate is 6% in the first year and 8% in the second year.

Solution:
A = P x (1 + R1/100) x (1 + R2/100)
A = 20,000 x (1 + 6/100) x (1 + 8/100)
A = 20,000 x 1.06 x 1.08
A = 20,000 x 1.1448
A = Rs 22,896

CI = 22,896 - 20,000 = Rs 2,896.

Answer: The amount is Rs 22,896 and the CI is Rs 2,896.

Example 6: Example 6: Half-yearly compounding

Problem: Find the CI on Rs 15,000 at 12% per annum for 1 year, compounded half-yearly.

Solution:
When compounded half-yearly: Rate = R/2 = 12/2 = 6% per half year. Time = 2n = 2 x 1 = 2 half-year periods.

A = P(1 + 6/100)^2 = 15,000(1.06)^2 = 15,000 x 1.1236 = Rs 16,854.

CI = 16,854 - 15,000 = Rs 1,854.

Note: If compounded annually: CI = 15,000 x 12/100 = Rs 1,800. Half-yearly compounding gives Rs 54 more.

Answer: The compound interest (half-yearly) is Rs 1,854.

Example 7: Example 7: Finding the rate of interest

Problem: Rs 6,250 amounts to Rs 7,290 in 2 years at compound interest. Find the rate.

Solution:
A = P(1 + R/100)^n
7,290 = 6,250(1 + R/100)^2
(1 + R/100)^2 = 7,290/6,250 = 1.1664
1 + R/100 = square root of 1.1664 = 1.08
R/100 = 0.08
R = 8%

Answer: The rate of interest is 8% per annum.

Example 8: Example 8: Population growth using CI formula

Problem: The population of a town is 50,000. It increases at 4% per annum. Find the population after 3 years.

Solution:
Using the CI formula for growth:
Population after n years = P(1 + R/100)^n
= 50,000(1 + 4/100)^3
= 50,000(1.04)^3
= 50,000 x 1.124864
= 56,243 (approximately)

Answer: The population after 3 years will be approximately 56,243.

Example 9: Example 9: Depreciation using CI formula

Problem: A car worth Rs 5,00,000 depreciates at 10% per annum. Find its value after 2 years.

Solution:
For depreciation, we use (1 - R/100) instead of (1 + R/100):
Value = P(1 - R/100)^n
= 5,00,000(1 - 10/100)^2
= 5,00,000(0.90)^2
= 5,00,000 x 0.81
= Rs 4,05,000

Answer: The value of the car after 2 years is Rs 4,05,000.

Example 10: Example 10: CI on a loan

Problem: Priya borrowed Rs 25,000 from a bank at 12% per annum compound interest. She repaid Rs 10,000 at the end of the first year. How much does she need to pay at the end of the second year to clear the loan?

Solution:
End of Year 1:
Amount = 25,000 x (1 + 12/100) = 25,000 x 1.12 = Rs 28,000.
After repaying Rs 10,000: Remaining = 28,000 - 10,000 = Rs 18,000.

End of Year 2:
Amount on remaining = 18,000 x 1.12 = Rs 20,160.

She needs to pay Rs 20,160 at the end of Year 2 to clear the loan.

Answer: Rs 20,160.

Real-World Applications

Compound interest has many real-world applications beyond simple savings:

Banking and Savings: Banks calculate interest on savings accounts, fixed deposits, and recurring deposits using compound interest. Understanding CI helps you choose the best savings plan.

Loans and EMIs: Home loans, car loans, and personal loans all use compound interest. The total amount you repay is calculated using CI formulas. EMI (Equated Monthly Instalment) calculations are based on compound interest.

Investment Growth: Mutual funds, stock markets, and other investments grow according to compound interest principles. Long-term investments benefit enormously from compounding.

Population Growth: The compound interest formula is used to project population growth. If a city grows at 3% per year, its future population can be estimated using the CI formula.

Depreciation: The value of vehicles, machinery, and electronics decreases over time. Depreciation is calculated using a modified CI formula with subtraction instead of addition.

Inflation: The rising cost of goods over time follows compound growth. If inflation is 5% per year, the price of an item costing Rs 100 today will be Rs 100 x (1.05)^n after n years.

Key Points to Remember

Compound interest is calculated on the principal plus previously accumulated interest — "interest on interest."
The amount formula is A = P(1 + R/100)^n, where P = principal, R = rate per annum, n = time in years.
CI = A - P = P[(1 + R/100)^n - 1].
For the first year, CI equals SI. The difference appears from the second year onwards.
CI - SI for 2 years = P(R/100)^2.
For half-yearly compounding, use rate = R/2 and time = 2n half-year periods.
For quarterly compounding, use rate = R/4 and time = 4n quarters.
When rates differ by year, multiply the individual growth factors.
For depreciation, use (1 - R/100) instead of (1 + R/100).
Compound interest always yields more than simple interest for more than one period at the same rate.

Practice Problems

Find the compound interest on Rs 12,000 at 10% per annum for 2 years.
What sum will amount to Rs 9,261 at 5% per annum compound interest in 3 years?
Find the difference between CI and SI on Rs 15,000 at 8% for 2 years.
Rs 40,000 is invested at 6% per annum compounded half-yearly. Find the amount after 1 and 1/2 years.
A machine worth Rs 2,00,000 depreciates at 15% per annum. Find its value after 3 years.
Find the CI on Rs 8,000 for 2 years at 5% for the first year and 10% for the second year.
The population of a village is 20,000 and it grows at 5% per annum. What will be the population after 2 years?
Rs 16,000 is borrowed at 12.5% per annum CI. Find the amount to be paid after 3 years.

Frequently Asked Questions

Q1. What is compound interest?

Compound interest is the interest calculated on the initial principal and also on the accumulated interest from previous periods. Unlike simple interest, where interest is only on the principal, compound interest earns 'interest on interest.'

Q2. What is the formula for compound interest?

The amount formula is A = P(1 + R/100)^n, where P is the principal, R is the rate per annum (%), and n is the time in years. Compound Interest = A - P.

Q3. How is CI different from SI?

In SI, interest is calculated only on the original principal. In CI, interest is calculated on the principal plus any interest already earned. For the same P, R, and time (more than 1 year), CI is always greater than SI.

Q4. What does 'compounded annually' mean?

Compounded annually means the interest is calculated once every year. At the end of each year, the interest is added to the principal, and the new principal is used for the next year's calculation.

Q5. How do you calculate CI compounded half-yearly?

For half-yearly compounding, divide the annual rate by 2 and multiply the time in years by 2. Use the formula: A = P(1 + R/200)^(2n).

Q6. Is CI always greater than SI?

For a time period of exactly one year (compounded annually), CI equals SI. For any period longer than one year, CI is greater than SI. The difference grows as the time period increases.

Q7. Can compound interest be used for depreciation?

Yes. For depreciation, use the formula A = P(1 - R/100)^n. The minus sign indicates that the value decreases over time instead of increasing.

Q8. What is the shortcut for CI minus SI for 2 years?

The difference CI - SI for 2 years = P x (R/100)^2. For example, for P = 10,000 and R = 5%, the difference = 10,000 x (0.05)^2 = 10,000 x 0.0025 = Rs 25.

Q9. What happens when interest is compounded more frequently?

The more frequently interest is compounded (quarterly > half-yearly > annually), the more total interest is earned. This is because interest gets added to the principal more often, creating a larger base sooner.

Q10. How is the CI formula used in population growth?

Population growth uses the same formula: Future Population = Present Population x (1 + Growth Rate/100)^n. The growth rate replaces the interest rate, and the population replaces the principal.