Class 10: Compound Interest

$\displaystyle \textbf{Compound Interest}$
$\displaystyle \text{In Compound Interest (C.I.), if the interest earned during a period is not paid, it is added to the}$
$\displaystyle \text{original principal. The new amount then becomes the principal for the next conversion period.}$
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$\displaystyle \text{The difference between the final amount and the original principal is called Compound Interest.}$
$\displaystyle \text{Compound Interest (C.I.) = Final Amount - Original Principal = A - P}$
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$\displaystyle \textbf{Conversion Period}$
$\displaystyle \text{The time interval after which the principal changes is called the conversion period.}$
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$\displaystyle \textbf{Examples}$
$\displaystyle \text{1. If interest is compounded annually, the principal changes every year.}$
$\displaystyle \text{2. If interest is compounded half-yearly, the principal changes every six months.}$
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$\displaystyle \textbf{Note:}$
$\displaystyle \text{In Simple Interest, the principal remains the same throughout the loan period.}$
$\displaystyle \text{In Compound Interest, the principal changes after every conversion period as specified}$
$\displaystyle \text{in the terms of the loan.}$
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$\displaystyle \textbf{Also Note:}$
$\displaystyle \text{For the same principal and the same rate of interest compounded annually,}$
$\displaystyle \text{C.I. of }3^{\text{rd}}\text{ Year}>\text{C.I. of }2^{\text{nd}}\text{ Year}>\text{C.I. of }1^{\text{st}}\text{ Year}.$
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$\displaystyle \text{Similarly, for the same principal and the same rate of interest compounded half-yearly,}$
$\displaystyle \text{C.I. of }3^{\text{rd}}\text{ Half-Year}>\text{C.I. of }2^{\text{nd}}\text{ Half-Year}>\text{C.I. of }1^{\text{st}}\text{ Half-Year}.$
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$\displaystyle \text{The difference between the Compound Interests of two consecutive conversion periods is the}$
$\displaystyle \text{interest of one conversion period on the Compound Interest of the preceding period.}$
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$\displaystyle \textbf{Example:}$
$\displaystyle \text{If }\text{Rs. }1000\text{ and }\text{Rs. }1050\text{ are the C.I. of two consecutive years, then}$
$\displaystyle \text{Rs. }1050-\text{Rs. }1000=\text{Rs. }50\text{ is the interest of one year on Rs. }1000.$
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$\displaystyle \text{Similarly, the difference between the amounts of two consecutive conversion periods is the}$
$\displaystyle \text{interest of one conversion period on the amount of the preceding period.}$
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$\displaystyle \textbf{Example:}$
$\displaystyle \text{If }\text{Rs. }1000\text{ and }\text{Rs. }1100\text{ are the amounts of two consecutive periods, then}$
$\displaystyle \text{Rs. }1100-\text{Rs. }1000=\text{Rs. }100\text{ is the interest on Rs. }1000.$
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$\displaystyle \textbf{Important Results}$
$\displaystyle \text{1. If the C.I. of the first conversion period is Rs. }x,\text{ then the C.I. of the next period is}$
$\displaystyle \text{Rs. }x+\text{interest for one period on Rs. }x.$
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$\displaystyle \text{2. If the amount at the end of a conversion period is Rs. }y,\text{ then the amount for the}$
$\displaystyle \text{next period is Rs. }y+\text{interest for one period on Rs. }y.$
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$\displaystyle \textbf{Formulae for Compound Interest}$
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$\displaystyle \text{1. When interest is compounded annually}$
$\displaystyle A=P\left(1+\frac{r}{100}\right)^n$
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$\displaystyle \text{2. When the rates of successive years are different}$
$\displaystyle A=P\left(1+\frac{r_1}{100}\right)\left(1+\frac{r_2}{100}\right)\left(1+\frac{r_3}{100}\right)\cdots$
$\displaystyle \text{where }r_1\%,r_2\%,r_3\%,\ldots\text{ are the rates for successive years.}$
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$\displaystyle \text{3. Compound Interest}$
$\displaystyle \text{C.I.}=A-P$
$\displaystyle \text{C.I.}=P\left(1+\frac{r}{100}\right)^n-P$
$\displaystyle \text{C.I.}=P\left[\left(1+\frac{r}{100}\right)^n-1\right]$
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$\displaystyle \text{4. When interest is compounded half-yearly}$
$\displaystyle A=P\left(1+\frac{r}{2\times100}\right)^{2n}$
$\displaystyle \text{Here, the rate is divided by }2\text{ and the number of years is multiplied by }2.$
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$\displaystyle \text{5. When interest is compounded quarterly}$
$\displaystyle A=P\left(1+\frac{r}{4\times100}\right)^{4n}$
$\displaystyle \text{Here, the rate is divided by }4\text{ and the number of years is multiplied by }4.$
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$\displaystyle \text{Application of the Compound Interest Formula}$
$\displaystyle \text{The compound interest formula is used to calculate the growth of quantities such as}$
$\displaystyle \text{population, production, industries, revenue, investments, number of trees, etc.}$
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$\displaystyle \text{If a quantity grows at }r\%\text{ per year, then after }n\text{ years,}$
$\displaystyle A=P\left(1+\frac{r}{100}\right)^n$
$\displaystyle \text{where }A\text{ is the final value, }P\text{ is the initial value and }n\text{ is the number of years.}$
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