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Chapter 1: Problem 70

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

Short Answer

Expert verified
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 by step solution

01

Express the Formula

Firstly express the formula with all variables: \(S=\frac{C}{1-r}\)
02

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: \(S(1 - r) = C\)
03

Expand the Left Side

Distribute \(S\) on the left side of the equation to remove the parenthesis. It will yield: \(S - Sr = C\)
04

Isolate the term containing \(r\)

Isolate \(Sr\) by subtracting \(S\) from both sides of the equation. That will result in: \(-Sr = C - S\)
05

Isolate \(r\)

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

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