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# How do you solve for $P$: $A = P(1 + rt)$?

Last updated date: 10th Aug 2024
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Hint:Here $P$ is on the Right Hand Side, i.e., RHS. Solving for $P$ here means rearranging the given equation such that $P$ is written in terms of all other variables in the equation. For this, $P$ is to be shifted to LHS and other variables to RHS such that $P$ becomes the dependent variable, the value of which depends on the other independent variables, i.e. $A$, $r$ and $t$.

Complete step by step solution:
Given equation is $A = P(1 + rt)$
Since P is on the RHS, we first interchange the terms of LHS and RHS. We get,
$P(1 + rt) = A$
Now, we divide both sides by $(1 + rt)$. Here we have to realise that the expression $(1 + rt)$ is a non-zero number, since dividing a number by zero gives an undefined result.
For case $(1 + rt) = 0$, $rt$ must be equal to $- 1$. For our discussion here we assume that $rt$ cannot be equal to $- 1$.
Since, $(1 + rt)$ cannot be equal to zero, we divide both sides by $(1 + rt)$.
$\Rightarrow \dfrac{{P(1 + rt)}}{{(1 + rt)}} = \dfrac{A}{{(1 + rt)}} \\ \Rightarrow P = \dfrac{A}{{(1 + rt)}} \\$
From this equation, we get $P$ in terms of the variables $A$, $r$ and $t$. Knowing the value of $A$, $r$ and $t$ we can solve the above equation to get the value of $P$.

Additional Information: The given equation is the formula of calculating amount $A$ to be repaid for principal $P$ with simple interest calculated at rate $(r \times 100)$ per annum after time $t$ years.

Note: If $rt = - 1$, $P$ becomes undefined as $A$ divided by $0$is undefined for any value of $A$. Therefore assumption of $rt \ne - 1$is to be held true to solve for $P$. Writing $P$ in terms of other variables means $P$ is the dependent variable which is dependent on $A$, $r$ and $t$ which are known as the independent variables.