Question

How do I find an interest rate using the formula $a = p{(1 + r)^t}$?

Answer 1

Hint: In the above question, we have a compound interest formula to find the interest rate using it, which is possible if we have the required values of amount, principal and time that is being taken to return the amount. For the formula above, we just need values of the required variables.

Complete step-by-step solution:
To find the interest rate $(r)$ in the formula $a = p{(1 + r)^t}$ , you need to know the values of \[a\] (amount), $p$ (principal) and $t$ (time). You would take \[a\] and divide it by $p$ . You will then take that result and take the $t$ root of it. You then subtract that answer by $1$ to get your interest rate in decimal form.
Here is an example:
You need to find the interest rate of an account that grew from \[\$ 4000\] to $\$ 4063$ in $2$ years.
Amount \[a = 4063\]
Principle \[p = 4000\]
Time $t = 2$
Placing the values in the required formulas,
$\Rightarrow$\[4063 = 4000{(1 + r)^2}\]
Now shifting the multiplied number to the left-hand side and dividing it,
$\Rightarrow$\[\dfrac{{4063}}{{4000}} = {(1 + r)^2}\]
Simplifying the equation,
$\Rightarrow$\[1.01575 = {(1 + r)^2}\]
Taking square roots on both the sides,
$\Rightarrow$\[\surd 1.01575 = \surd {(1 + r)^2}\]
The values after taking the square roots are,
$\Rightarrow$\[1.007844 = 1 + r\]
Shifting constants on the same side and performing subtraction on it,
$\Rightarrow$\[1.007844 - 1 = r\]
The final value of the interest rate is,
$\Rightarrow$\[0.007844 = r\]
Interest rate rounds to \[.784\% \]
We can find interest rates as shown above.

Note: Interest rate is actually the amount that is charged, expressed as a percentage of the principal, by a lender to a borrower for the use of assets. It is the interest due per period as a proportion of the amount lent, deposited or borrowed, which is also called as the principal sum. Typically, the amount of interest rate is charged annually, known as the annual percentage rate APR.