Simple & Compound Interest

Interest is extra money calculated on an amount. Simple interest uses the starting amount; compound interest also includes earlier interest.

Match rate and number of periods. Compound multipliers multiply; simple interest adds equal amounts based on the original principal.

₹100 earns 10% interest each year for 2 years, with annual compounding. Find the final amount. 1. After year one: ₹100 + ₹10 = ₹110. 2. Year two interest is 10% of ₹110, which is ₹11. 3. The final amount is ₹121.

The second interest calculation uses ₹110, which includes first-year interest. This produces ₹121 after two years, compared with ₹120 under simple interest at the same annual rate. The diagram assumes annual compounding and no deposits or withdrawals; different periods need a matching rate and number of periods.

At a nominal annual rate of 12%, compounded half-yearly for 18 months, there are 3 periods at 6%. On 10000, amount=10000 x 1.06^3=11910.16; interest=1910.16. Simple interest at 12% for 1.5 years would be 1800. The difference comes from interest earning interest in later periods, not from a longer time.

The compounding period matters. The second year does not always use the original amount. Do not confuse nominal annual rate with effective annual rate. These are mathematical examples, not loan or investment advice.

An annual rate cannot be used unchanged as a monthly rate. The rate and time period must refer to the same interval.

Match rate and number of periods. Compound multipliers multiply; simple interest adds equal amounts based on the original principal.

At a nominal annual rate of 12%, compounded half-yearly for 18 months, there are 3 periods at 6%. On 10000, amount=10000 x 1.06^3=11910.16; interest=1910.16. Simple interest at 12% for 1.5 years would be 1800. The difference comes from interest earning interest in later periods, not from a longer time.

Do not confuse nominal annual rate with effective annual rate. These are mathematical examples, not loan or investment advice.

With principal P, annual percentage rate r and time t in years, simple interest I=Prt/100 and amount A=P+I. For a nominal annual rate r compounded m times per year for an integer number mt of periods, A=P(1+r/(100m))^(mt).

Annual compounding uses m=1, half-yearly m=2 and quarterly m=4. For different period rates, multiply the corresponding growth factors. Compound interest equals A−P. With annual growth fraction a=r/100, the difference CI−SI is Pa² over two years and Pa²(3+a) over three years, assuming identical principal and rate.

On 1000 at 10% annually for two years, SI=200 and the simple-interest amount is 1200. Annual compounding gives A=1000×1.1²=1210 and CI=210. At a nominal 10% compounded half-yearly for one year, A=1000×1.05²=1102.50. Growth times If simple interest makes an amount X times the principal in T years, the time to become Y times is T(Y−1)/(X−1) when the denominator is nonzero. For compound growth at a constant positive rate, it is T log(Y)/log(X), when the model and log domains apply. Population growth uses the same multiplicative model only under its stated constant-rate assumption.

Compounding means adding interest to the balance at the specified intervals; a withdrawal changing the balance is not itself compounding. Fractional periods require the stated convention, such as simple interest for the leftover fraction or fractional-power growth. Do not silently choose between them.