Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Write an equation that represents the statement. The quotient of \(r\) and 2 is 9 .

Step-by-Step Solution

Answer

The equation is \(\frac{r}{2} = 9\).

Step 1: Understand the statement

The statement says 'The quotient of r and 2 is 9'. This means r divided by 2 equals 9.

Step 2: Formulate the equation

Based on the understanding from Step 1, the equation can be written as \(\frac{r}{2} = 9\).

Key Concepts

Quotient

When tackling algebraic equations, understanding the word "quotient" is essential. A quotient is the result of dividing one number by another. For example, in our exercise, the term "quotient" tells us that the problem involves division. Specifically, we are dividing a variable, denoted by \(r\), by the number 2.

• In mathematical terms, if you divide a number \(a\) by another number \(b\), the result is \(\frac{a}{b}\), which is the quotient.
• A clear understanding of this concept translates directly to setting up correct equations, which is crucial in algebra.

Remember, when you see the word "quotient," think about the division operation. In our problem, it specifically means that \(\frac{r}{2}\) is the expression needed for further calculations.

Equation Formulation

Equation formulation is the process of translating a word problem into a mathematical statement. In our case, we transformed "The quotient of \(r\) and 2 is 9" into an equation. It is crucial to correctly interpret the language. Here’s how to break it down:

• First, identify the components involved: the variable \(r\), the constant 2, and the result 9.
• Translate the word "quotient" into the corresponding mathematical operation, which is division in this context.
• Structure this understanding into the equation \(\frac{r}{2} = 9\).

Start by identifying keywords in the problem like "quotient" or "equals"—these guide you in constructing the equation effectively. Formulating equations correctly is essential for problem-solving in algebra. It provides a clear model to follow and solve.

Problem Solving

Problem solving involves using the formulated equation to find the value of the unknown variable. With the equation \(\frac{r}{2} = 9\), you need to isolate \(r\). This process requires operations that maintain the equation's balance.

• In this case, multiply both sides of the equation by 2 to undo the division. This gives us \(r = 18\).
• It's important to ensure every operation applied to one side of the equation is also applied to the other side to keep it equal.
• Once \(r\) is isolated, you've solved the equation!

Effective problem solving in algebra is about taking systematic steps to find solutions. Be methodical: carefully manipulate the equation until the variable is isolated. This builds confidence and competence in handling algebraic equations.