Question

The formula A=P(1+r)t is used to show the total amount owed for a loan with a simple annual interest rate.

Answer 1

Answer:  $r=(\frac{A}{P})^{\frac{1}{t}}-1$

Step-by-step explanation

Compound interest is the addition of interest to the principal sum of a deposit or a loan.

Let P = principal amount which was taken as a loan then the accumulated amount A is given by

$A=P(1+r)^t$.......(1)

where, r is the rate of simple annual interest in decimal.

t is the time applied for interest.

For solving r divide both sides of equation by P in (1),we get

$\frac{A}{P}=(1+r)^t\\\Rightarrow(\frac{A}{P})^{\frac{1}{t}}=1+r\\\Rightarrow\ r=(\frac{A}{P})^{\frac{1}{t}}-1$.

Answer 2

We are asked to express r in terms of A, P, and t.

We first divide both sides of the equation by t, which gives us

                                         
                     $\displaystyle{ \frac{A}{t}=P(1+r)$, 


then, dividing both sides by P, we have

                     $\displaystyle{ \frac{A}{Pt}=1+r$.

Swap the sides:

                    $\displaystyle{ 1+r= \frac{A}{Pt}$ 
      
Finally subtracting 1 from both sides gives us

                     $\displaystyle{ r=\frac{A}{Pt}-1$.