Problem 70 - College Algebra Essentials

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$S=\frac{C}{1-r} \text { for } r$$

Step-by-Step Solution

Answer

The formula solved for \(r\) is: \(r= \frac{S - C}{S}\). The original formula is a geometric series sum formula, where \(S\) is the sum of the series, \(C\) is the first term and \(r\) is the common ratio.

Step 1: Express the Formula

Firstly express the formula with all variables: \(S=\frac{C}{1-r}\)

Step 2: Cross-Multiply

The goal is to clear the denominator on the right side of the equation. That can be achieved by multiplying both sides of the equation by \(1-r\), leading to: \(S(1 - r) = C\)

Step 3: Expand the Left Side

Distribute \(S\) on the left side of the equation to remove the parenthesis. It will yield: \(S - Sr = C\)

Step 4: Isolate the term containing \(r\)

Isolate \(Sr\) by subtracting \(S\) from both sides of the equation. That will result in: \(-Sr = C - S\)

Step 5: Isolate \(r\)

Finally, isolate \(r\) by dividing every term in the equation by \(-S\). The final formula for \(r\) is: \(r= \frac{S - C}{S}\)