PDF chapter test

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-yearly.

The 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 quarterly.

The 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:

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

And $$C.I$$ $$=$$ Amount $$A$$ $$-$$ Principal $$P$$.