Question

How much would $\mathrm{\$}150$ invested at $8\mathrm{\%}$ interest compounded continuously be worth after 17 years?

Answer 1

Hint: Here, we will substitute the given values in the formula of the amount. Then we will simplify it further to find the required amount after 17 years. The amount is the money which is the summation of the principal amount and interest accumulated on the principal at a certain rate for a fixed period of time.

Formula Used:
$A=P{\left(1+{\displaystyle \frac{r}{100}}\right)}^n$
Where, $A=$ Final amount
$P=$ Principal
$r=$ Rate of interest per annum
$n=$ Number of years

Complete step-by-step answer:
Compound Interest is basically the addition of interest in the Principal amount or in simple terms we can say, reinvesting of interest.
Formula of Compound interest is: $A=P{\left(1+{\displaystyle \frac{r}{100}}\right)}^n$
Now, according to the question,
A sum of money (i.e. Principal) is $\mathrm{\$}150$ which is invested at compounded interest (which means we have to use the above formula) at $8\mathrm{\%}$ per annum for 17 years.
Hence, we have to find the total amount $A$
Number of years $=17$
Using the formula, we get,
$A=150{\left(1+{\displaystyle \frac{8}{100}}\right)}^{17}$
$\Rightarrow A=150{\left(1+0.08\right)}^{17}=150{\left(1.08\right)}^{17}$
Hence, we get,
$\Rightarrow A=150\left(3.7\right)=555$
Therefore, the amount after 17 years is $\mathrm{\$}555$

Hence, if $\mathrm{\$}150$ is invested at $8\mathrm{\%}$ interest compounded continuously then its worth after 17 years will be $\mathrm{\$}555$.
Thus, this is the required answer.

Note:
In this question we have used the formula of Compound Interest. Compound Interest is calculated both on the Principal as well as on the accumulated interest of the previous year. Hence, this is also known as ‘interest on interest’.
Its formula is:
$C.I=P{\left(1+{\displaystyle \frac{R}{100}}\right)}^n-P$
Where, $C.I$ is the Compound Interest, $P$ is the Principal, $R$ is the rate of interest per annum and $n$ is the time period.
But, in order to calculate the amount, we used the formula:
$A=P{\left(1+{\displaystyle \frac{r}{100}}\right)}^n$
The second type of interest is Simple Interest. Simple Interest is the interest earned on the Principal or the amount of loan. Its formula is, as we have discussed,
$S.I={\displaystyle \frac{P.R.T}{100}}$
Where, $S.I$ is the Simple Interest, $P$ is the Principal, $R$ is the rate of interest per annum and $T$ is the time period.