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We can calculate the compound interest easily by using the formula. But those formula differs from the method of using compound interest in real life. Now we will take a look at those types to understand the compound interest more clear.

Whatever the question is, first, we have to calculate the amount of the respective year at a particular rate of interest and principal. Using the amount, we can easily calculate the compound interest.

*Type 1*: General formula to find the compound interest and the amount.

The formula to calculate the amount is given by $A\phantom{\rule{0.147em}{0ex}}=P{\left(1+\frac{r}{100}\right)}^{n}$.

Here \(A\) is the

**amount**,\(r\) is the**rate of interest per annum**, and \(n\) is the**time period**.Now using this amount value, we can determine the compound interest \(C.I\) as follows:

\(C.I\) \(=\) Amount \(A\) \(-\) Principal \(P\)

We can also use the alternative formula, which is obtained by combining both the formula above.

\(C.I\) \(=\) \(A\) \(-\) \(P\)

$C.I\phantom{\rule{0.147em}{0ex}}=\left[P{\left(1+\frac{r}{100}\right)}^{n}\right]-P$

Factor out \(P\) from the right-hand side of the equation.

Therefore, $C.I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}P\left[{\left(1+\frac{r}{100}\right)}^{n}-1\right]$.

*Type 2*: To find the amount and the compound interest when compounded annually or half-yearly or quarterly.

Let \(A\) be the

**amount**, \(P\) be the**principal**, \(r\) be the**rate of interest per annum**, and \(n\) be the**time period**.**:**

*Case 1**To find the amount and the compound interest when compounded annually*.

The formula for calculating the amount when compounded annually is given by $A\phantom{\rule{0.147em}{0ex}}=P{\left(1+\frac{r}{100}\right)}^{n}$.

And \(C.I\) \(=\) Amount \(A\) \(-\) Principal \(P\).

Or $C.I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}P\left[{\left(1+\frac{r}{100}\right)}^{n}-1\right]$.

*.*

**Case 2**: To find the amount and the compound interest when compounded half-yearlyThe formula for calculating the amount when compounded half-yearly is given by $A\phantom{\rule{0.147em}{0ex}}=P{\left(1+\frac{r}{200}\right)}^{2n}$.

And \(C.I\) \(=\) Amount \(A\) \(-\) Principal \(P\).

Or $C.I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}P\left[{\left(1+\frac{r}{200}\right)}^{2n}-1\right]$.

*.*

**Case 3**: To find the amount and the compound interest when compounded quarterlyThe formula for calculating the amount when compounded quarterly is given by $A\phantom{\rule{0.147em}{0ex}}=P{\left(1+\frac{r}{400}\right)}^{4n}$.

And \(C.I\) \(=\) Amount \(A\) \(-\) Principal \(P\).

Or $C.I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}P\left[{\left(1+\frac{r}{400}\right)}^{4n}-1\right]$.

*Type 3*: To find the amount when the interest is compounded annually but the rate of interest differs year by year.

Let \(A\) be the

**amount**, \(P\) be the**principal**, \(r_{1}, r_{2}, r_{3}\) be the**interest rates for first, second and third consecutive years per annum**and \(n\) be the**time period**.Then the amount at the end of \(n\) years is given by:

$A\phantom{\rule{0.147em}{0ex}}=P\left(1+\frac{{r}_{1}}{100}\right)\left(1+\frac{{r}_{2}}{100}\right)\left(1+\frac{{r}_{3}}{100}\right).......\left(1+\frac{{r}_{n}}{100}\right)$

And \(C.I\) \(=\) Amount \(A\) \(-\) Principal \(P\).

*Type 4*: To find the amount when interest is compounded annually but time being a fraction.

Let \(A\) be the

**amount**, \(P\) be the**principal**, \(r\) be the**rate of interest per annum**and \(n=\) $a\frac{b}{c}\phantom{\rule{0.147em}{0ex}}\mathit{years}$ be the**time period**.Then the amount at the end of \(n\) years is given by:

$A\phantom{\rule{0.147em}{0ex}}=P\phantom{\rule{0.147em}{0ex}}{\left(1+\frac{r}{100}\right)}^{a}\left(1+\frac{r\times \frac{b}{c}}{100}\right)$

And \(C.I\) \(=\) Amount \(A\) \(-\) Principal \(P\).