# An amount of AUD500 is invested in a bank at compound interest?

## (a) at 6% p.a. at quarterly rests for $3$ years, then what is the interest earned? (b) In another bank, the interest received is same but at an annually compounding rate, then what is the interest rate paid?

Jun 17, 2017

(a) Interest earned is $A U D$ $97.81$

(b) Second bank offers 6.14% p.a.

#### Explanation:

If an amount $P$ is invested at a simple rate of interest $r$ for a period of $t$ years

the total amount becomes $P {\left(1 + \frac{r}{100}\right)}^{t}$, if interest is compounded annually.

if it is compounded more frequently, say $n$ times in a year, the amount becomes $P {\left(1 + \frac{r}{100 \times n}\right)}^{n \times t}$

As here, interest is compounded quarterly at 6% amount
of $A U D 500$ in three years becomes

$500 {\left(1 + \frac{6}{100 \times 4}\right)}^{4 \times 3} = 500 \times {1.015}^{12}$

= $500 \times 1.19562 = A U D \text{ } 597.81$ i.e. interest is $A U D$ $97.81$

Now let us say he earns a rate of r_2% p.a. in second bank, compounded annually.

Then as the amount would be $597.81 A U D$, we have

$500 \times {\left(1 + {r}_{2} / 100\right)}^{3} = 597.81$

or ${\left(1 + {r}_{2} / 100\right)}^{3} = \frac{597.81}{500} = 1.19562$

or $1 + {r}_{2} / 100 = \sqrt{1.19562} = 1.061364$

and r_2=0.061364xx100=6.14% p.a. (upto $2 \mathrm{dp}$)