Question

How much would $\$400$ invested at $9\%$ interest compounded continuously be worth after $3$ years?

Answer 1

Hint: As the given rate of interest is given as compound interest, so for calculating the value of the amount which is to be obtained after the end of $3$ years, apply the formula of amount in case of compound interest.
$A=P{{\left[ 1+\left( \dfrac{R}{100} \right) \right]}^{t}}$
where, $A$ is the value of Amount which we get after the end of $3$ years
$P$ is the value of amount which is invested
$R$ is the percentage rate on which the money is invested
t is the value of the time period for which the principal or the money is invested.
with the help of the above given formula determine the worth of $\$400$ after $3$ year.

Complete step by step solution:
As per data given in the question,
As here the rate of interest is compounded in nature,
As per question,
We have,
Principal $=P=\$400$
Rate percent $=9\%$
Time period $=3$ years
So,
Here, the value of Amount will be calculating with the help of formula of Compound interest,
So,
Applying the formula,
As we know that,
$A=P{{\left[ 1+\left( \dfrac{R}{100} \right) \right]}^{t}}$
Where,
$P$ is the amount which is to be lent or principal
$R$ will be value of rate percent
$T$ will be the time period.
So,
$A=400{{\left[ 1+\left( \dfrac{9}{100} \right) \right]}^{t}}$
$\Rightarrow A=400\times {{\left[ \dfrac{\left( 100+9 \right)}{100} \right]}^{3}}$
$\Rightarrow A=400\times \dfrac{109}{100}\times \dfrac{109}{100}\times \dfrac{109}{100}$
$\Rightarrow A=\$518.0116$
Hence, value of amount after $3$ years with the rate of $9\%$ will be $\$518.0116$

Additional information:
While calculating the value of interest,
There are major fours terms are used,
Principal – it is the value of amount which is lent or given to any person
Rate - it is the value of percentage of rate for which the amount is given to a certain person/
Time – it is the value of time period for which the amount or the money is given to any person.
On basis of rate and for the time which the amount is given,
The value of interest is calculated.
Amount – it is a value which any person who takes a certain money for a certain period will return back to the person from whom he took after addition of the money which he took and the value of interest charged.
So, we can say that,
The value of Amount is always and always more than the value of principal.
There are basically two types of interest,
One is simple interest while the other one is compound interest
In simple interest we calculate the rate of interest on the primary amount.
Like, if Rs. $100$ is lent for $2$ years at $20\%$ of simple interest,
Then the person has to pay $10\%$ of the principal i.e. Rs. $10$ for each and every year, it means to say that the amount of interest will be equal for each and every year.
But in case of compound interest a person is charged in the successive interest way.
Means here in case of compound interest the interest is calculated on the principal + interest at end of the year.
Like, if Rs. $100$ is lent for $2$ years at $10\%$ of interest,
Then, in first year he has to pay $10\%$ of 100 i.e. Rs.$10$
But in second year he has to pay $10\%$ of (principal + interest at the end of first year)
So, interest for second year will be $10\%$ of $\left( 100+10 \right)$
$=10\%$ of $110=$ Rs. $11$

Note: The value of simple interest and the value of compound interest for first years is always equal.
Here, in question as the Way of charged interest is given as compound so while calculating the value of Amount apply only the formula of compound interest not of the formula of calculating the interest of simple interest.