Problem 31 - Algebra A Combined Function

Question

Solve the following. One-fourth equals the quotient of a number and 8 . Find the number.

Step-by-Step Solution

Answer

The number is 2.

Step 1: Understand the Equation

The problem states that one-fourth is equal to the quotient of a number and 8. To write an equation for this, let the unknown number be represented by \( x \). Thus, we have the equation: \[ \frac{1}{4} = \frac{x}{8} \].

Step 2: Cross-Multiply

To solve for \( x \), we need to eliminate the fractions by cross-multiplying. Thus, multiply both sides by 8, the denominator of the fraction on the right side: \[ 1 \cdot 8 = 4 \cdot x \].

Step 3: Simplify the Equation

After cross-multiplying, the equation becomes: \( 8 = 4x \). Now, we need to solve for \( x \).

Step 4: Solve for x

Divide both sides of the equation by 4 to isolate \( x \). Thus, \[ x = \frac{8}{4} \].

Step 5: Simplify the Result

Simplify the division to find the value of \( x \). \( \frac{8}{4} \) simplifies to \( 2 \). Thus, \( x = 2 \).

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding the value of an unknown variable that makes the equation true. An equation is essentially a balance between two expressions, and solving it requires performing equal operations on both sides to maintain that balance. In this particular problem, the basic algebraic equation is presented as \( \frac{1}{4} = \frac{x}{8} \), where solving for \( x \) is our goal. Understanding the structure of the equation and the relationships between its components helps in deciding the steps needed to isolate \( x \) and find the solution. The critical part of solving such equations lies in manipulating them without altering the equality.

Fractions

Fractions represent parts of a whole and are expressed as one number over another, separated by a line, such as \( \frac{1}{4} \) or \( \frac{x}{8} \). In the context of equations, fractions can sometimes complicate the process of solving for a variable. However, they can be managed through simple algebraic techniques.
Understanding fractions requires knowing their components: the numerator (the top number) and the denominator (the bottom number), which indicates how many parts the whole is divided into. In our equation, both sides feature fractions, and understanding how to manipulate them is crucial to finding the variable. Often, cross multiplication or multiplying both sides by the denominator can help eliminate fractions, simplifying the equation.

Cross Multiplication

Cross multiplication is a powerful technique used to solve equations involving fractions. This method allows us to eliminate the fractions and solve for the unknown variable more easily. In the equation \( \frac{1}{4} = \frac{x}{8} \), cross multiplication involves multiplying the numerator of one fraction by the denominator of the other.

• First, multiply the left numerator (1) by the right denominator (8).
• Then, multiply the right numerator (x) by the left denominator (4).

This gives the equation \( 1 \cdot 8 = 4 \cdot x \). Cross multiplication turns an equation of two fractions into a straightforward linear equation, making it more manageable to solve. By doing so, we maintain the balance of the equation and can proceed to further simplify to find the solution.

Math Problem Solving

Math problem solving is an essential skill that involves more than just performing calculations. It's about understanding the problem, applying the correct methods, and verifying the solution. The process is methodical and requires critical thinking. In this exercise, problem-solving began by translating a word problem into an algebraic equation.
Once the equation \( \frac{1}{4} = \frac{x}{8} \) was set up, recognizing that cross-multiplication would lead us to the solution was key. After solving for \( x \) through division, verifying that \( x = 2 \) satisfies the original equation confirms the accuracy of our solution.
Effective problem solving also involves reflecting on the steps taken, ensuring that each step logically follows from the previous one and contributes towards finding the correct answer.