Question

How much would $\$500$ invested at 6% interest compounded monthly be worth after years, $A\left( t \right) = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$?

Answer 1

Hint:
In this question we are asked to find the amount after years when some principle is invested at a rate, we will solve this by using the formula $A\left( t \right) = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$, where Principal amount invested P, r rate of interest per annum and n tells us how frequently (at regular intervals) interest is compounded in a year, and this gives the amount at the end of t years. By substituting the values in the formula that are given in the question, we will get the required amount.

Complete step by step solution:
Given Principal amount invested P, r rate of interest per annum and n tells us how frequently (at regular intervals) interest is compounded in a year, and this gives the amount at the end of t years, i.e., $A\left( t \right) = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$where P is the principle , r is the rate,
Now from the given data, $P = \$ 500$, $r = 6\% = \dfrac{6}{{100}}$and $n = 12$as it is compounded monthly,
By substituting the values in the formula we get,
$ \Rightarrow A\left( t \right) = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$,
Now substituting the values we get,
$ \Rightarrow A\left( t \right) = 500{\left( {1 + \dfrac{{\dfrac{6}{{100}}}}{{12}}} \right)^{12t}}$
Now simplifying we get,
$ \Rightarrow A\left( t \right) = 500{\left( {1 + \dfrac{6}{{1200}}} \right)^{12t}}$,
Now again simplifying we get,
$ \Rightarrow A\left( t \right) = 500{\left( {1 + \dfrac{1}{{200}}} \right)^{12t}}$,
Now adding the terms inside the power we get,
$ \Rightarrow A\left( t \right) = 500{\left( {\dfrac{{200 + 1}}{{200}}} \right)^{12t}}$,
Now simplifying we get,
$ \Rightarrow A\left( t \right) = 500{\left( {\dfrac{{201}}{{200}}} \right)^{12t}}$,
So, amount when $\$500$ invested at 6% interest compounded monthly be worth after years is $500{\left( {\dfrac{{201}}{{200}}} \right)^{12t}}$.

$\therefore $ The amount when $\$500$ invested at 6% interest compounded monthly be worth after years will be equal to $500{\left( {\dfrac{{201}}{{200}}} \right)^{12t}}$.

Note:
Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is different from the simple interest where interest is not added to the principal while calculating the interest during the next period.