Problem 13 - Fundamentals of Mathematics

Question

Determine the missing numerator or denominator. (Section 4.4) \(\frac{4}{3}=\frac{32}{?}\)

Step-by-Step Solution

Answer

The missing denominator is 24.

Step 1: Set up the equation

Since \( \frac{4}{3} = \frac{32}{?} \), we need to determine the missing denominator. Let the missing denominator be \( x \). So, we can rewrite the equation as \( \frac{4}{3} = \frac{32}{x} \).

Step 2: Cross-multiply to form an equation

Cross-multiplying the fractions gives us \( 4 \times x = 32 \times 3 \). This results in the equation \( 4x = 96 \).

Step 3: Solve for the missing denominator

To solve for \( x \), divide both sides of the equation \( 4x = 96 \) by 4. This gives us \( x = \frac{96}{4} \), so \( x = 24 \).

Key Concepts

Numerator

In a fraction, the numerator plays a crucial role. It is the number that sits above the line in a fraction, representing the part of the whole we are considering. For example, in the fraction \( \frac{4}{3} \), 4 is the numerator. It tells us "four parts of something divided into three parts."
Understanding the Numerator:

• The numerator indicates how many parts of the unit fraction are being considered.
• If the numerator is equal to the denominator, the fraction equals 1.
• A larger numerator compared to the denominator indicates an improper fraction, representing more than a whole unit.

In our original problem, the given fraction is \( \frac{32}{x} \). Here, 32 is the numerator. We need to determine what number \( x \) is, so the two fractions are equal. The numerator of a fraction provides valuable insight into the size or quantity of the portion we are considering.

Denominator

The denominator is another fundamental part of a fraction. It is the number found below the line and indicates the total number of equal parts into which the whole is divided. For example, in the fraction \( \frac{4}{3} \), the number 3 is the denominator.
Understanding the Denominator:

• The denominator shows the total number of equal parts the whole is divided into.
• A larger denominator means each piece is smaller, as the whole is divided into more parts.
• A denominator should not be zero, as division by zero is undefined in mathematics.

In the problem we are addressing, we initially have an unknown denominator represented by \( x \) in \( \frac{32}{x} \). To find this missing denominator, it is crucial to understand the role it plays in making fractions equivalent.

Cross-Multiplication

Cross-multiplication is a technique often used to solve equations involving fractions. This method helps us find an unknown in a proportion by multiplying across the equal sign diagonally.
Steps of Cross-Multiplication:

• Identify the two fractions that make up the equation, such as \( \frac{4}{3} = \frac{32}{x} \).
• Multiply the numerator of the first fraction by the denominator of the second fraction (4 \( \times \) \( x \)).
• Then, multiply the denominator of the first fraction by the numerator of the second fraction (3 \( \times \) 32).
• Set the two results equal to each other: \( 4x = 96 \).

By solving this equation, we find the value of \( x \), which in this case, turned out to be 24 after simplifying. Cross-multiplication is a straightforward and powerful technique to quickly determine unknowns in proportion problems.