Problem 79 - 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. $$\frac{3}{a}=\frac{?}{a^{2}}$$

Step-by-Step Solution

Answer

The expression is \(3a\).

Step 1: Understand the Problem

We need to find an expression to replace the question mark so that the two rational expressions are equivalent: \(\frac{3}{a} = \frac{?}{a^2}\). This means the two fractions should be equal in value.

Step 2: Rewrite the Equations

The expressions are given as: \(\frac{3}{a}\) and \(\frac{?}{a^2}\). To compare these, note that for fractions to be equivalent, the cross-products of the numerator and denominator must be the same.

Step 3: Set Up the Cross-Multiplication

For fraction equivalence, set up the cross-multiplication equation: \(3 \times a^2 = ? \times a\).

Step 4: Solve for the Question Mark

To find the value of the question mark (\(?\)), solve the equation \(3a^2 = ?a\). Divide both sides by \(a\) to isolate the question mark: \(\frac{3a^2}{a} = ?\). Simplify the result: \(3a = ?\).

Step 5: Final Expression

Therefore, the expression that replaces the question mark is \(3a\).

Key Concepts

equivalent fractions

In mathematics, when two fractions are equivalent, they represent the same value, even though they may look different. For instance, \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent because if you multiply the numerator and the denominator of \(\frac{1}{2}\) by 2, you get \(\frac{2}{4}\). Similarly, if you simplify \(\frac{2}{4}\) by dividing both the numerator and the denominator by 2, you get \(\frac{1}{2}\).

To determine if the rational expressions \(\frac{3}{a}\) and \(\frac{?}{a^2}\) are equivalent, we need to find a value for the question mark so that both fractions have the same value.

One useful technique for verifying equivalence of fractions is cross-multiplication, which we will explore in the next section.

cross-multiplication

Cross-multiplication is a useful technique for determining if two fractions are equivalent. It involves multiplying the numerator of one fraction by the denominator of the other fraction and comparing the results.

In our problem, we have the fractions \(\frac{3}{a}\) and \(\frac{?}{a^2}\). To check their equivalence, we set up a cross-multiplication equation:

\[3 \times a^2 = ? \times a\]

This equation helps us compare the cross-products of the numerators and denominators.

By solving this equation for the question mark (?), we can find the value that makes the two fractions equivalent. We do this by isolating the question mark on one side of the equation.

simplifying algebraic expressions

Simplifying algebraic expressions means reducing them to their simplest form. This often involves factoring, combining like terms, and canceling common factors.

In our exercise, we have the cross-multiplication equation:

\[3a^2 = ?a\]

To isolate the question mark (?), we divide both sides of the equation by \(a\):

\[\frac{3a^2}{a} = ?\]

Simplifying the left side of the equation, we get:

\[3a = ?\]

Therefore, the expression that replaces the question mark is \(3a\).

By understanding these core concepts—equivalent fractions, cross-multiplication, and simplifying algebraic expressions—we can solve rational expressions effectively and ensure we arrive at the correct solution.