Simple and Compound Interest

Simple and Compound Interest

It is the sum which is paid by the borrower to the lender for using the money for a specific time period. The money borrowed is the principal. The rate at which the interest is calculated on the Principal is called Rate of Interest. The time for which the money is borrowed is the Time and the total sum of principal and interest is called the Amount.

SIMPLE INTEREST (SI)

When interest is calculated on borrowed money thus interest is known as simple interest. Simple interest is denoted by SI.

\[\text{SI=}\frac{\text{Principal }\!\!\times\!\!\text{ Time }\!\!\times\!\!\text{ Rate}}{\text{100}}\text{=}\frac{\text{P }\!\!\times\!\!\text{ t }\!\!\times\!\!\text{ r}}{\text{100}}\]

\[\text{SI=}\frac{\text{Art}}{\text{100+rt}};\] where A = Amount; r = Rate; t = Time

\[\text{A=P+SI};\] where A = Amount; P = Principal;

            SI = Simple Interest

\[\text{P=}\frac{\text{100 }\!\!\times\!\!\text{ A}}{\text{100+rt}}\text{;}\] where A = Amount; r = Rate; t = Time

AMOUNT

The sum of interest and principal is known as amount.

            Amount = Principal + SI/CI

COMPOUND INTEREST (CI)

Interest that accrues on the initial principal and the

Accumulated interest of a prime debt is called compound interest. Compounding of interest allows a principal amount to grow at a faster rate.

If compound interest accrued on Rs. P at r% rate of compound interest for T yr and compound amount is Rs. A, then

            \[A=P{{\left( 1+\frac{r}{100} \right)}^{T}}\]

            \[CI=P\left\{ {{\left( 1+\frac{r}{100} \right)}^{T}}-1 \right\}\]

            \[r=\left\{ {{\left( \frac{A}{P} \right)}^{\frac{1}{T}}}-1 \right\}\times 100%\]

When the rate of interest for three consecutive years are \[{{r}_{1}}%,\,\,{{r}_{2}}%\] and \[{{r}_{3}}%\] respectively, then

            \[A=P\left( 1+\frac{{{r}_{1}}}{100} \right)\left( 1+\frac{{{r}_{2}}}{100} \right)\left( 1+\frac{{{r}_{3}}}{100} \right)\]

·         If interest is compounded half-yearly, then

Amount \[=P{{\left( 1+\frac{R}{2\times 100} \right)}^{2n}}\]

·         If interest is compounded quarterly, then

Amount \[=P{{\left( 1+\frac{R}{4\times 100} \right)}^{4n}}\]

·         If interest is compounded annually but time is in fraction (Suppose, time \[=t\frac{a}{b}yr\]), then

Amount \[=P{{\left( 1+\frac{R}{100} \right)}^{t}}\times \left( 1+\frac{(a/b)\,R}{100} \right)\]

Difference in Interest

As the time passes on, principal for CI calculation increases, resulting more interest as compare to SI, hence the existence of difference between compound interest and simple interest comes in picture and getting larger with the time-flow as interest is directly proportional to the principal, so go through the difference calculation for various year given below.

Difference between CI and SI for One Year

There is no difference between CI and SI for 1 yr as the principal in both cases remain same resulting 0 difference, hence \[CI-SI=0\]

Difference between CI and SI for Two Years

There is a little difference between CI and SI for 2 yr as principal in CI is affected with addition of first year interest, resulting greater principal as compare to the principal for SI calculation. Hence, \[CI-SI=\frac{P\times {{r}^{2}}}{{{(100)}^{2}}}\]

Difference between CI and SI for Three Years

There is little difference between CI and SI for 3 yr and is given by \[CI-SI=\frac{P\times {{r}^{2}}\times (300+r)}{{{(100)}^{3}}}\]

Quicker One

Ø   If an amount becomes \[{{x}_{1}}\] times of itself in \[{{n}_{1}}\] yr and \[{{x}_{2}}\] times of itself in \[{{n}_{2}}\] yr.

Ø   If a sum of money becomes Rs. \[{{x}_{1}}\] after \[{{n}_{1}}\] yr at the rate of compound interest and Rs. \[{{x}_{2}}\] after \[{{n}_{2}}\] yr, then rate of interest,

\[r=\left\{ {{\left( \frac{{{x}_{2}}}{{{x}_{1}}} \right)}^{\frac{1}{{{n}_{2}}-{{n}_{1}}}}}-1 \right\}\times 100%\]

Ø   If compound interest on an amount for 2 yr is CI and simple interest on that money is SI, then principal \[=\frac{{{(SI)}^{2}}}{4\,\,(CI-SI)}\]

Ø   If compound interest on an amount for two consecutive years are \[{{C}_{1}}\] and \[{{C}_{2}}\] respectively, then rate of interest \[=\frac{{{C}_{2}}-{{C}_{1}}}{{{C}_{1}}}\times 100%\]

Ø   If a sum of money becomes x times at the rate of compound interest in n yr, then it will take \[y\times n\] yr to become \[{{x}^{y}}\] times.

Ø   If a certain sum, at compound interest becomes x times in \[{{T}_{1}}\] yr and y times in \[{{T}_{2}}\] yr, then \[{{(x)}^{\frac{1}{{{T}_{1}}}}}={{(y)}^{\frac{1}{{{T}_{2}}}}}.\]