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, fracP(-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 fracPcancel(-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 = -fracA-1 - rt. Step 5.3.2: Rewrite -1 as its equivalent, -1(1), to get P = -fracA-1(1) - rt. Step 5.3.3: Factor out -1 from -rt, resulting in P = -fracA-1(1) - (rt). Step 5.3.4: Factor -1 from the entire denominator to obtain P = -fracA-1(1 + rt). Step 5.3.5: Finally, simplify the expression. Step 5.3.5.1: Move the negative sign in front, yielding P = --fracA1 + rt. Step 5.3.5.2: Multiply -1 by -1 to get P = 1fracA1 + rt. Step 5.3.5.3: Multiply the …

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

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Solution:

Step 1:

Eliminate $P r t$ from both sides of the equation to get $A - P - P r t = 0$ .

Step 2:

Move $A$ to the right side, resulting in $- P - P r t = - A$ .

Step 3:

Extract the common factor $P$ from the terms on the left side.

Step 3.1:

Take $P$ out of $- P$ , yielding $P (- 1) - P r t = - A$ .

Step 3.2:

Continue factoring $P$ from $- P r t$ , which gives $P (- 1) + P (- r t) = - A$ .

Step 3.3:

Combine the factored terms to form $P (- 1 - r t) = - A$ .

Step 4:

Simplify the term $- 1 r$ to $- r$ , resulting in $P (- 1 - r t) = - A$ .

Step 5:

Divide the equation by $- 1 - r t$ to isolate $P$ .

Step 5.1:

Perform the division, $\frac{P (- 1 - r t)}{- 1 - r t} = \frac{- A}{- 1 - r t}$ .

Step 5.2:

Simplify the left side by canceling out the common factors.

Step 5.2.1:

Cancel the $- 1 - r t$ term to get $\frac{P (- 1 - r t)}{(- 1 - r t)} = \frac{- A}{- 1 - r t}$ .

Step 5.2.1.1:

After canceling, you're left with $P = \frac{- A}{- 1 - r t}$ .

Step 5.3:

Simplify the right side of the equation.

Step 5.3.1:

Bring the negative sign to the front, giving $P = - \frac{A}{- 1 - r t}$ .

Step 5.3.2:

Rewrite $- 1$ as its equivalent, $- 1 (1)$ , to get $P = - \frac{A}{- 1 (1) - r t}$ .

Step 5.3.3:

Factor out $- 1$ from $- r t$ , resulting in $P = - \frac{A}{- 1 (1) - (r t)}$ .

Step 5.3.4:

Factor $- 1$ from the entire denominator to obtain $P = - \frac{A}{- 1 (1 + r t)}$ .

Step 5.3.5:

Finally, simplify the expression.

Step 5.3.5.1:

Move the negative sign in front, yielding $P = - - \frac{A}{1 + r t}$ .

Step 5.3.5.2:

Multiply $- 1$ by $- 1$ to get $P = 1 \frac{A}{1 + r t}$ .

Step 5.3.5.3:

Multiply the fraction by $1$ , resulting in $P = \frac{A}{1 + r t}$ .

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.