Problem 78 - Algebra for College Students

Question

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{3}{4}=\frac{12}{?}$$

Step-by-Step Solution

Answer

The missing denominator is 16.

Step 1: Identify the relationship between numerators

First, notice the relationship between the numerators of the two rational expressions. The numerator of the second fraction is 12, while the numerator of the first fraction is 3. Determine how to transform 3 into 12.

Step 2: Find the multiplier

To find the multiplier, divide 12 by 3: \(\frac{12}{3} = 4\). This indicates that the numerator is being multiplied by 4 to go from the first fraction to the second.

Step 3: Apply the same multiplier to the denominator

Since the fractions are equivalent, the same multiplier must be applied to the denominator of the first fraction. Multiply the denominator 4 by 4: \(4 \times 4 = 16\).

Step 4: Fill in the missing denominator

The denominator of the second fraction must be 16. Therefore, \(\frac{3}{4} = \frac{12}{16}\).

Key Concepts

numerator and denominator relationship

When working with rational expressions, it is crucial to understand how the numerator (top number) and the denominator (bottom number) relate to each other. In our example, we have two fractions: \(\frac{3}{4}\) and \(\frac{12}{?}\). We first look at the numerators, which are 3 and 12. By examining these terms, we realize that 12 is a multiple of 3. This relationship shows us that we need a specific number, called a multiplier, to transform 3 into 12.

Similarly, the relationship between the denominators is connected. Once we figure out the multiplier for the numerators, we apply the same multiplier to the denominator. This ensures that the fractions stay equivalent. It helps to remember that any change in the numerator must be mirrored by an equivalent change in the denominator to maintain the same ratio.

multipliers

Multipliers play a critical role in creating equivalent fractions. In our example, we determine the multiplier for the numerators first. If we divide 12 by 3, we get 4. This means that to transform \(\frac{3}{4}\) into \(\frac{12}{?}\), the numerator 3 is multiplied by 4:

\[ 3 \times 4 = 12 \]

Next, since the fractions are equivalent, we need to multiply the denominator by the same multiplier. Here, the original denominator is 4:

\[ 4 \times 4 = 16 \]

This calculation shows that the missing denominator is 16. Therefore, we can say:

\(\frac{3}{4} = \frac{12}{16}\)

Understanding multipliers helps you make transformations in both numerators and denominators, which is central for solving rational expressions.

equivalent fractions

Equivalent fractions are different fractions that represent the same value. In our exercise, \(\frac{3}{4}\) is equivalent to \(\frac{12}{16}\) because they represent the same portion of a whole. This is achieved by multiplying both the numerator and the denominator by the same number, known as the multiplier.

Key points to remember about equivalent fractions:

• They always have the same ratio or value.
• If you multiply or divide both the numerator and the denominator by the same number, the fractions remain equivalent.
• This property helps with simplifying fractions or finding common denominators in addition and subtraction.

In practice, identifying and creating equivalent fractions is essential for solving more complex mathematical problems, including algebraic equations and proportional reasoning.