The text below is just to refresh your memories. You could also click on the link above to revise what you learned on Compound Interest in Class 8. You might also want to look at some of the exercises that you solved on this topic in Class 8 (Lecture Notes / Exercise 15A / Exercise 15B).

Principal: This is the money borrowed or lent out for a certain period of time is called the principal or sum.

Interest: Interest is payment from a borrower to a lender of an amount above repayment of the principal sum

Amount: The total money paid back by the borrower to the lender is called the amount.

Rate: The interest on Rs. 100 for a unit time is called the rate of interest. It is expressed in %. The interest on Rs. 100 for 1 year is called rate % per annum (abbreviated as rate % p. a.) $\displaystyle \text{Amount = Principal + Interest}$

Simple Interest

• Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains. It excludes the effect of compounding. It is denoted by S.I.
• The simple interest is calculated uniformly only on the original principal throughout the loan period.
• You do not earn interest on the interest earned during the loan period.

Formula: Let Principal = P, Rate = R% per annum and Time T years. Then, we have: $\displaystyle S.I. = \frac{P \times R \times T}{100}$

Notes:

• While calculating the time period between two given dates, the day on which the money is borrowed is not counted for interest calculations while the day on which the money is returned, is counted for interest calculations.
• For converting the time in days into years, we always divide by 365, whether it is an ordinary year or a leap year.

Compound Interest

• Compound interest includes interest earned on the interest which was previously accumulated.
• Here you also earn interest over the interest accrued during the loan period.
• The difference between the final amount and the original principal is called the compound interest (abbreviated as C.I.) $\displaystyle \text{Compound Interest (C.I.) = Final Amount - Original Principal}$

Computation of compound interest

Type I: Computing interest when interest is compounded annually.

• Step 1: In this case just use the formula: $\displaystyle \text{Interest } = \frac{P \times R \times T}{100} \text{ for the 1st year. }$

• Step 2: Add this amount to the Starting principal of the 1st year to get the starting principal for the 2nd year.
• Step 3: Calculate the interest for the 2nd year using the same formula just that the principal is now what you calculated in step 2.
• Step 4: Repeat the above steps if you have more number of years for which the interest has to be calculated.

Type II: Computing interest when interest is compounded half-yearly.

• In a scenario like this, if the rate of interest is $\displaystyle R\%$ per annum, and the interest is compounded half-yearly, then the rate of interest will be $\displaystyle \frac{R}{2} \%$ per half-year.
• The logic of calculation remains the same as explained above, except that the rate becomes half.
• Here you calculate the interest for 1st half-year, 2nd half year, and so on.

Type III: Computing interest when interest is compounded quarterly.

• In a scenario like this, if the rate of interest is $\displaystyle R\%$ per annum, and the interest is compounded quarterly, then the rate of interest will be $\displaystyle \frac{R}{4} \%$ per half year.

Computation of compound interest using formula

Theorem 1: Let Principal $\displaystyle = P$, Rate of interest $\displaystyle = R\%$ per annum. If the interest is compounded annually, then the amount A and compound interest (C.I.) at the end of $\displaystyle n$ years is given by: $\displaystyle A = P \Big( 1+ \frac{R}{100} \Big)^n \text{ and } C.I. = A - P = P \Big[ \Big(1+ \frac{R}{100} \Big)^n - 1 \Big]$

Theorem 2: Let Principal $\displaystyle = P$, Rate of interest $\displaystyle = R\%$ per annum. If the interest is recognized k times a yes, then the amount A and compound interest (C.I.) at the end of $\displaystyle n$ years is given by: $\displaystyle A = P \Big( 1+ \frac{R}{100k} \Big)^{nk} \text{ and } C.I. = A - P = P \Big[ \Big(1+ \frac{R}{100k} \Big)^{nk} - 1 \Big]$

Theorem 3: Let Principal $\displaystyle = P$, Rate of interest $\displaystyle = R_1\%$ for the first year, $\displaystyle = R_2\%$ for the second year and $\displaystyle = R_3\%$ for the third year and so on and the last $\displaystyle = R_n\%$ for the $\displaystyle n^{th}$ year. Then the amount A and compound interest (C.I.) at the end of $\displaystyle n$ years is given by: $\displaystyle A = P \Big(1+ \frac{R_1}{100} \Big).\Big(1+ \frac{R_2}{100} \Big).\Big(1+ \frac{R_3}{100} \Big)....\Big(1+ \frac{R_n}{100} \Big) \text{ and } C.I. = A - P$

Theorem 4: Let Principal $\displaystyle = P$, Rate of interest $\displaystyle = R\%$ per annum. If the interest is compounded annually, but the time is in fraction of year, say $\displaystyle 3 \frac{1}{4}$  years, then the amount A is given by: $\displaystyle A = P \Big(1+ \frac{R}{100} \Big)^3 \Big(1 + \frac{\frac{R}{4}}{100} \Big)$

Formula for Population Growth

Theorem 1: If $\displaystyle P$ is the population of the city / town at the beginning of a certain year and the population grows at $\displaystyle R\%$ per annum, then $\displaystyle \text{Population after } n \text{ Year } = P \Big(1+ \frac{R}{100} \Big)^{n}$

Theorem 2: If $\displaystyle P$ is the population of the city / town at the beginning of a certain year and the population grows at $\displaystyle R_1\%$ during the first year, $\displaystyle R_2\%$ during the second year, $\displaystyle R_3\%$ during the third year, and so on… $\displaystyle \text{Population after } 3^{rd} \text{ Year } = P \Big(1+ \frac{R_1}{100} \Big).\Big(1+ \frac{R_2}{100} \Big).\Big(1+ \frac{R_3}{100} \Big)$

Theorem 3: If $\displaystyle P$ is the population of the city / town at the beginning of a certain year and the population decreases at $\displaystyle R\%$ per annum, then $\displaystyle \text{Population after } n \text{ years } = P \Big(1- \frac{R}{100} \Big)^{n}$

Depreciation

Theorem 1: If $\displaystyle V_0$ is the value of an equipment at a certain time and $\displaystyle R\%$ per annum is the rate of depreciation, then the value $\displaystyle V_n$ at the end of $\displaystyle n$ years is given by: $\displaystyle V_n = V_0 \Big( 1- \frac{R}{100} \Big)^n$

Theorem 2: If $\displaystyle V_0$ is the value of an equipment at a certain time and the rate of depreciation for first $\displaystyle n_1$ years is $\displaystyle R_1\%$, for second $\displaystyle n_2$ year is $\displaystyle R_2\%$, for third $\displaystyle n_3$ years is $\displaystyle R_3\%$ and so on and $\displaystyle R_n\%$ for the last $\displaystyle n_k$ year, then the value at the end of $\displaystyle n_1+n_2+n_3+...+n_k$ years is given by: $\displaystyle V_n = V_0 \Big(1- \frac{R_1}{100} \Big)^{n_1}.\Big(1- \frac{R_2}{100} \Big)^{n_2}.\Big(1- \frac{R_3}{100} \Big)^{n_3}....\Big(1- \frac{R_n}{100} \Big)^{n_k}$