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Chapter 2: Problem 95

Solve each formula for the specified variable. \(M=C(1+r)\) for \(r\)

Short Answer

Expert verified
r = \frac{M}{C} - 1

Step by step solution

01

Understand the given equation

The given equation is \[ M = C(1 + r) \]We need to solve this equation for the variable \( r \).
02

Isolate the term containing the variable

First, divide both sides of the equation by \( C \), the coefficient that is multiplying \( (1 + r) \):\[ \frac{M}{C} = \frac{C(1 + r)}{C} \]This simplifies to:\[ \frac{M}{C} = 1 + r \]
03

Solve for the variable

To isolate \( r \), subtract 1 from both sides of the equation:\[ \frac{M}{C} - 1 = r \]This gives:\[ r = \frac{M}{C} - 1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
Algebraic equations are mathematical statements that use variables to represent unknown values. These equations often include operations such as addition, subtraction, multiplication, and division. Solving algebraic equations typically involves finding the value of the variable that makes the equation true. The variable is usually represented by letters like x, y, or r.
For example, in the equation \( M = C(1 + r) \), M, C, and r are variables. Given that we need to solve for r, we have to manipulate the equation to isolate r on one side of the equation.
Isolating Variables
Isolating variables is a crucial step in solving algebraic equations. It means getting the variable you are solving for on one side of the equation and everything else on the other side.
  • First, identify the term that contains the variable.
  • Perform operations to isolate the term containing the variable.
  • Lastly, solve for the variable.
In our example, the equation is \( M = C(1 + r) \). To isolate r, follow these steps:
  • Divide both sides by C: \[ \frac{M}{C} = \frac{C(1 + r)}{C} \]
  • Simplify to get: \[ \frac{M}{C} = 1 + r \]
  • Subtract 1 from both sides: \[ \frac{M}{C} - 1 = r \]
Thus, r has been isolated and solved for: \[ r = \frac{M}{C} - 1 \]
Fraction Manipulation
Fraction manipulation involves basic arithmetic operations with fractions, such as addition, subtraction, multiplication, and division. Understanding how to work with fractions is essential in solving algebraic equations, especially when isolating variables. Here are some key operations for manipulating fractions:
  • **Multiplication**: Multiply both the numerator (the top number) and the denominator (the bottom number).
  • **Division**: Divide by flipping the second fraction and then multiplying.
  • **Addition and Subtraction**: Make sure the fractions have a common denominator before performing the operation.
In our equation, \( M = C(1 + r) \), we used division and subtraction to isolate r.
Dividing both sides of the equation by C and then subtracting 1 required basic fraction operations. By understanding these steps, you can solve many algebraic equations with ease.

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Most popular questions from this chapter

If 2 is added to five times a number, the result is equal to 5 more than four times the number. Find the number.

If 3 is added to a number and this sum is doubled, the result is 2 more than the number. Find the number.

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Solve each equation for \(y\) $$ -3 x+5 y=-15 $$
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