Problem 76 - 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. $$4=\frac{?}{3}$$

Step-by-Step Solution

Answer

The expression is 12.

Step 1 - Understand the Problem

The goal is to find a numerator that, when divided by 3, equals 4.

Step 2 - Set Up the Equation

We need to find the numerator, so let's call it 'x'. We set up the equation: \[ 4 = \frac{x}{3} \]

Step 3 - Solve for x

Multiply both sides of the equation by 3 to isolate x: \[ 4 \times 3 = x \] \[ x = 12 \]

Step 4 - Substitute Back

Substitute 12 back into the fraction to ensure it equals 4: \[ \frac{12}{3} = 4 \] This confirms that 12 is the correct numerator.

Key Concepts

Numerator

The numerator is a key part of any fraction. It is the number situated above the line in a fraction and it signifies how many parts of the whole are being considered. For instance, in the fraction \(\frac{3}{5}\), the numerator is 3. This tells us that 3 parts of a total of 5 are taken into account.
In the provided exercise, we are asked to find the numerator that, when divided by 3, equals 4. Imagine you're splitting something into 3 parts, and then you want the total to equal 4. By multiplying 4 by 3, you get 12. So, 12 must be the numerator. Hence, \(\frac{12}{3}\) is equal to 4.

Equivalent Fractions

Equivalent fractions represent the same value, even though they may look different. For example, \(\frac{1}{2}\) is equivalent to \(\frac{2}{4}\) or \(\frac{3}{6}\). They simplify to the same value, which is 0.5 in this case.
To determine if \(\frac{12}{3}\) is equivalent to 4, the key is understanding that multiplying the numerator by or dividing it by a number should not alter the value of the fraction. Both fractions should simplify to the same final value.
In the exercise provided, \(\frac{12}{3}=4\), demonstrating our fraction is equivalent to 4 as simplifying \(\frac{12}{3}\) yields the integer 4. This confirms that 12 is indeed the correct numerator.

Basic Algebra

Basic algebra involves manipulating equations to find unknown variables. Here, we're tasked with finding an expression that makes two sides of an equation equivalent. In our exercise, we set the problem as \[ 4 = \frac{x}{3} \]
Solving for \('x'\):

• We multiply both sides of the equation by 3 to isolate x: \(\[\begin{equation} \[ 4 \times 3 = x \] \[ x = 12 \] \end{equation}\]\)
• Then, substitute 12 back into the fraction for verification: \(\[\begin{equation} \[ \frac{12}{3} = 4 \] \end{equation}\]\)

This comfortably leads us to the correct answer, which is crucial in understanding the relationship between the division and multiplication processes in algebra. By breaking down the equation step by step, algebra helps us find the right numerator needed to make the original statement true.