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What is the difference between the compound interests on $Rs.5000$ for $1\frac{1}{2}$ years at $4$% per annum compounded yearly and half-yearly?

See we know formula for compound interest $P\left ( 1+\frac{R}{100} \right )^{n}$

But when they calculating compound interest for $Rs.5000$ they calculated like this $5000\left ( 1+\frac{4}{100} \right )\times \left ( 1+\frac{\frac{1}{2}\times 4}{100} \right )$

Do we need to calculate fraction power like this?

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$5000\left ( 1+ \frac{4}{100} \right )^{1\tfrac{1}{2}}\;= 5000\left ( 1+ \frac{4}{100} \right )^{1+\tfrac{1}{2}}\;=5000\left ( 1+ \frac{4}{100} \right )^{1} \left ( 1+ \frac{4}{100} \right )^{1/2}$

Now, According to Binomial Approximation,

$5000\left ( 1+ \frac{4}{100} \right )^{1\tfrac{1}{2}}\;= 5000\left ( 1+ \frac{4}{100} \right )^{1} \left ( 1+ \frac{4*\frac{1}{2}}{100} \right )$

$P\left (1+ \frac{r}{100} \right )^n$ is a standard formula for calculating compound amount for n years.

Here they asked for compound interest for $1\tfrac{1}{2}$ years so first, they calculate the amount for 1st year which is $\left (1+ \frac{r}{100} \right )$.

Now for the remaining half year, they apply Rate of interest for only for half year i.e for a whole year if Rate of interest is 4% then for half year Rate of interest is $(\frac{4}{2})$%=2% and apply formula $\left (1+ \frac{r/2}{100} \right )$.

So Amount = $P\left (1+ \frac{r}{100} \right )$$\left (1+ \frac{r/2}{100} \right )$.

Also, you can directly apply the formula Amount = $P\left (1+ \frac{r/2}{100} \right )^3$.

Here we calculate the Compound amount for a half year so hear for  $1\tfrac{1}{2}$ we have a total 3 half years. Obviously, Rate of Interest is given in annum so for half yearly Rate of interest =$\tfrac{r}{2}$.
by Junior (561 points)

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