Class 9: Compound Interest (with formula)

$\displaystyle \textbf{Chapter 3: Compound Interest (Using Formula)}$

$\displaystyle \textbf{Concept Notes}$
$\displaystyle \textbf{1. Compound Interest (C.I.)}$
$\displaystyle \text{Compound Interest is the interest calculated on the original principal} \\ \text{as well as on the interest earned in previous years. Since interest is added } \\ \text{to the principal after every compounding period, the principal keeps increasing.}$

$\displaystyle \textbf{2. Important Terms}$
$\displaystyle \begin{array}{|c|l|} \hline \textbf{Term} & \textbf{Meaning} \\ \hline P & \text{Principal (Original Sum)} \\ \hline A & \text{Amount after Compound Interest} \\ \hline \text{C.I.} & \text{Compound Interest} \\ \hline r\% & \text{Rate of Interest per annum} \\ \hline n & \text{Number of years} \\ \hline \end{array}$

$\displaystyle \textbf{3. Formula for Annual Compounding}$
$\displaystyle A=P\left(1+\frac{r}{100}\right)^n$
$\displaystyle \text{where }P=\text{Principal},\ A=\text{Amount},\ r=\text{Rate of Interest},\ n=\text{Time in years.}$

$\displaystyle \textbf{4. Formula for Compound Interest}$
$\displaystyle \boxed{\text{C.I.}=A-P}$

$\displaystyle \textbf{5. Formula for Successive Different Rates}$
$\displaystyle A=P\left(1+\frac{r_1}{100}\right)\left(1+\frac{r_2}{100}\right)\left(1+\frac{r_3}{100}\right)\cdots$
$\displaystyle \text{where }r_1,r_2,r_3,\ldots\text{ are the rates of interest for successive years.}$

$\displaystyle \textbf{6. To Find the Principal}$
$\displaystyle P=\frac{A}{\left(1+\frac{r}{100}\right)^n}$

$\displaystyle \textbf{7. To Find the Rate of Interest}$
$\displaystyle \text{Substitute the given values in }A=P\left(1+\frac{r}{100}\right)^n\text{ and solve for }r.$

$\displaystyle \textbf{8. To Find the Time}$
$\displaystyle \text{Substitute the given values in }A=P\left(1+\frac{r}{100}\right)^n\text{ and solve for }n.$

$\displaystyle \textbf{9. Compound Interest Compounded Half-Yearly}$
$\displaystyle \text{For half-yearly compounding:}$
$\displaystyle \bullet\ \text{Rate of interest}=\frac{r}{2}\%$
$\displaystyle \bullet\ \text{Number of compounding periods}=2n$
$\displaystyle A=P\left(1+\frac{r}{2\times100}\right)^{2n}$

$\displaystyle \textbf{10. Compound Interest Compounded Quarterly}$
$\displaystyle \text{For quarterly compounding:}$
$\displaystyle \bullet\ \text{Rate of interest}=\frac{r}{4}\%$
$\displaystyle \bullet\ \text{Number of compounding periods}=4n$
$\displaystyle A=P\left(1+\frac{r}{4\times100}\right)^{4n}$

$\displaystyle \textbf{11. Fractional Number of Years}$
$\displaystyle \text{When the time is not an exact number of years and the interest is compounded yearly,} \\ \text{calculate the amount for the complete years first and then calculate the remaining part separately.}$
$\displaystyle \text{For }2\frac12\text{ years: }A=P\left(1+\frac{r}{100}\right)^2\left(1+\frac{r}{2\times100}\right)$

$\displaystyle \textbf{12. Compound Interest and Simple Interest}$
$\displaystyle \bullet\ \text{For one year, Compound Interest = Simple Interest.}$
$\displaystyle \bullet\ \text{For more than one year, Compound Interest > Simple Interest.}$

$\displaystyle \textbf{13. Growth Formula}$
$\displaystyle \text{The compound interest formula can also be used for population growth, industrial production,} \\ \text{growth of plants, inflation, etc.}$
$\displaystyle \text{Future Value}=\text{Present Value}\left(1+\frac{r}{100}\right)^n$

$\displaystyle \textbf{14. Depreciation Formula}$
$\displaystyle \text{When the value decreases every year by }r\%\text{:}$
$\displaystyle \text{Value after }n\text{ years}=P\left(1-\frac{r}{100}\right)^n$

$\displaystyle \textbf{15. Population Growth}$
$\displaystyle \text{Population after }n\text{ years}=\text{Present Population}\left(1+\frac{r}{100}\right)^n$
$\displaystyle \text{Present Population}=\frac{\text{Future Population}}{\left(1+\frac{r}{100}\right)^n}$

$\displaystyle \textbf{16. Industrial Production}$
$\displaystyle \text{Production after }n\text{ years}=\text{Present Production}\left(1+\frac{r}{100}\right)^n$

$\displaystyle \textbf{17. Important Formulae at a Glance}$
$\displaystyle \begin{array}{|l|} \hline A=P\left(1+\frac{r}{100}\right)^n\\ \hline \text{C.I.}=A-P\\ \hline P=\dfrac{A}{\left(1+\frac{r}{100}\right)^n}\\ \hline A=P\left(1+\frac{r}{2\times100}\right)^{2n}\quad\text{(Half-yearly)}\\ \hline A=P\left(1+\frac{r}{4\times100}\right)^{4n}\quad\text{(Quarterly)}\\ \hline \text{Growth}=P\left(1+\frac{r}{100}\right)^n\\ \hline \text{Depreciation}=P\left(1-\frac{r}{100}\right)^n\\ \hline \end{array}$

$\displaystyle \textbf{18. Examination Tips}$
$\displaystyle \bullet\ \text{Identify whether the interest is compounded yearly, half-yearly or quarterly.}$
$\displaystyle \bullet\ \text{For half-yearly compounding, divide the rate by }2\text{ and multiply the time by }2.$
$\displaystyle \bullet\ \text{For quarterly compounding, divide the rate by }4\text{ and multiply the time by }4.$
$\displaystyle \bullet\ \text{Always calculate the Amount first and then use }\text{C.I.}=A-P.$
$\displaystyle \bullet\ \text{For successive years with different rates, multiply the growth factors separately.}$
$\displaystyle \bullet\ \text{For depreciation problems, use }\left(1-\frac{r}{100}\right)\text{ instead of }\left(1+\frac{r}{100}\right).$
$\displaystyle \bullet\ \text{Read the question carefully to identify whether Principal, Amount, Compound Interest,} \\ \text{Rate or Time is to be found.}$