Solve for P A-P=Prt
The given problem is a basic algebraic equation where P represents the principal amount of money, r represents the rate of interest per time period, and t represents time. The equation A - P = Prt is typically used to calculate the interest earned over a given period of time when the total amount A at the end includes both the principal and the interest. The goal of the problem is to isolate and solve for the variable P (the principal amount) by manipulating the equation algebraically.
$A - P = P r t$
Answer
Solution:
Step 1:
Eliminate $Prt$ from both sides of the equation to get $A - P - Prt = 0$.
Step 2:
Move $A$ to the right side, resulting in $-P - Prt = -A$.
Step 3:
Extract the common factor $P$ from the terms on the left side.
Step 3.1:
Take $P$ out of $-P$, yielding $P(-1) - Prt = -A$.
Step 3.2:
Continue factoring $P$ from $-Prt$, which gives $P(-1) + P(-rt) = -A$.
Step 3.3:
Combine the factored terms to form $P(-1 - rt) = -A$.
Step 4:
Simplify the term $-1r$ to $-r$, resulting in $P(-1 - rt) = -A$.
Step 5:
Divide the equation by $-1 - rt$ to isolate $P$.
Step 5.1:
Perform the division, $\frac{P(-1 - rt)}{-1 - rt} = \frac{-A}{-1 - rt}$.
Step 5.2:
Simplify the left side by canceling out the common factors.
Step 5.2.1:
Cancel the $-1 - rt$ term to get $\frac{P\cancel{(-1 - rt)}}{\cancel{(-1 - rt)}} = \frac{-A}{-1 - rt}$.
Step 5.2.1.1:
After canceling, you're left with $P = \frac{-A}{-1 - rt}$.
Step 5.3:
Simplify the right side of the equation.
Step 5.3.1:
Bring the negative sign to the front, giving $P = -\frac{A}{-1 - rt}$.
Step 5.3.2:
Rewrite $-1$ as its equivalent, $-1(1)$, to get $P = -\frac{A}{-1(1) - rt}$.
Step 5.3.3:
Factor out $-1$ from $-rt$, resulting in $P = -\frac{A}{-1(1) - (rt)}$.
Step 5.3.4:
Factor $-1$ from the entire denominator to obtain $P = -\frac{A}{-1(1 + rt)}$.
Step 5.3.5:
Finally, simplify the expression.
Step 5.3.5.1:
Move the negative sign in front, yielding $P = --\frac{A}{1 + rt}$.
Step 5.3.5.2:
Multiply $-1$ by $-1$ to get $P = 1\frac{A}{1 + rt}$.
Step 5.3.5.3:
Multiply the fraction by $1$, resulting in $P = \frac{A}{1 + rt}$.
Knowledge Notes:
The problem involves solving a linear equation in one variable, where the variable $P$ is multiplied by a combination of constants and another variable. The steps include:
Isolating the variable: The first step in solving for $P$ is to isolate it on one side of the equation. This involves moving terms that do not contain $P$ to the other side.
Factoring: Factoring is a process where you take out a common factor from terms. In this case, $P$ is the common factor that is factored out to simplify the equation.
Simplification: After factoring, the equation is simplified by canceling out common terms or by combining like terms.
Division: To isolate $P$, the equation is divided by the coefficient of $P$ (which, after factoring, is a binomial expression).
Negative signs and simplification: Careful attention is given to the negative signs during simplification, especially when they appear in the denominator of a fraction. Factoring out a negative sign can often simplify the expression and make it easier to understand.
Final expression: The final step is to express $P$ in its simplest form, which involves ensuring that all negative signs are correctly placed and that the expression is as straightforward as possible.
This problem is a common type of algebraic manipulation that requires knowledge of basic algebraic operations, including distributing, factoring, and simplifying expressions.
