### Theory:

We can calculate the compound interest easily by using 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 calculate the amount of respective year at a particular rate of interest and principal then using the amount we can easily calculate the compound interest.
Type 1)

General formula:

To find the compound interest and the amount, we can use the following formula.

Step 1: First the find amount.

Amount $$A$$ $=P{\left(1+\frac{r}{100}\right)}^{n}$

Here $$A$$ be the amount, $$r$$ be the rate of interest per annum and $$n$$ be the time period.

Step 2: Now using this value of amount, we can determine the compound interest $$C.I$$ .

Compound interest ($$C.I$$) $$=$$ Amount ($$A$$) $$–$$ Principle ($$P$$)

We can also use alternative formula by combining the both formula above that is,

$\begin{array}{l}C.I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}A\phantom{\rule{0.147em}{0ex}}-P\\ \\ C.I\phantom{\rule{0.147em}{0ex}}=\left(P{\left(1+\frac{r}{100}\right)}^{n}\right)-P\end{array}$

Commonly taking out the $$P$$, then we will get,

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

Type 2)
Method to the amount and the compound interest when compounded annually or half yearly or quarterly.
Case 1)
To find the amount and the compound interest when compounded annually.
If $$A$$ be the amount, $$P$$ be the principal, $$r$$ be the rate of interest per annum and $$n$$ be the time period.

We can use the following formula for this condition that is,

Amount $$A$$ $=P{\left(1+\frac{r}{100}\right)}^{n}$

Or we can use below formula to find the $$C.I$$

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

Case 2)
To find the amount and the compound interest when compounded half - yearly.
If $$A$$ be the amount, $$P$$ be the principal, $$r$$ be the rate of interest per annum and $$n$$ be the time period.

We can use the following formula for this condition that is,

Amount $$A$$ $=P{\left(1+\frac{r}{200}\right)}^{2n}$

And $$C.I$$ will be $$=$$ $$A – P$$

Or we can use below formula to find $$C.I$$

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

Case 3)
To find the amount and the compound interest when compounded quarterly.
If $$A$$ be the amount, $$P$$ be the principal, $$r$$ be the rate of interest per annum and $$n$$ be the time period.

We can use the following formula for this condition that is,

Amount $$A$$ $=P{\left(1+\frac{r}{400}\right)}^{4n}$

And $$C.I$$ will be, $$C.I = A – P$$

Or we can use below formula to find $$C.I$$

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

Type 3)
To find the amount when the interest is compounded annually but rate of interest differs year by year.
If $$A$$ be the amount, $$P$$ be the principal, $$r1, r2, r3$$ be the interest rates for first, second and third consecutive years per annum and $$n$$ be the time period.

We can use the following formula for this condition that is,

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

Type 4)

Calculation of the amount when interest is compounded annually but time being a fraction.

If $$A$$ be the amount, $$P$$ be the principal, $$r$$ be the rate of interest per annum and $$n =$$ $a\frac{b}{c}$ years be the time period.

Therefore the formula will be:
Amount at the end of the $$n$$ years $$A$$ $=P\phantom{\rule{0.147em}{0ex}}{\left(1+\frac{r}{100}\right)}^{a}\left(1+\frac{r×\frac{b}{c}}{100}\right)$