Problem 89 - 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. $$\frac{3 a+3}{3 a}=\frac{?}{a}$$

Step-by-Step Solution

Answer

The expression is \(a + 1\).

Step 1 - Simplify the Left-Hand Side

First, simplify the left-hand side of the equation \(\frac{3a + 3}{3a}\). Factor out a common factor of 3 from the numerator: \(\frac{3(a + 1)}{3a}\).

Step 2 - Reduce the Fraction

Next, divide the numerator and the denominator by 3: \(\frac{3(a + 1)}{3a} = \frac{a + 1}{a}\). This simplifies the left-hand side to \(\frac{a + 1}{a}\).

Step 3 - Set the Expressions Equal

Now set this simplified expression equal to the right-hand side: \(\frac{a + 1}{a} = \frac{?}{a}\).

Step 4 - Determine the Missing Expression

Since both sides have the same denominator \(a\), the numerators must be equal. Therefore, the missing expression in the numerator must be \(a + 1\).

Key Concepts

simplifying rational expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Simplifying rational expressions involves making these fractions as simple as possible.
Here's how to simplify the following rational expression: \(\frac{3a + 3}{3a}\).
First, factor the numerator. In this case, we can take out the common factor 3: \frac{3(a + 1)}{3a}\.
When simplifying, always look for common factors in the numerator and the denominator.
Next, divide both by common factors, here we divide by 3: \frac{3(a + 1)}{3a} = \frac{a + 1}{a}\. This is the simplified form.

• Step 1 involves looking at both parts of the fraction and identifying common factors.
• Step 2 is about dividing those common factors out to simplify it.
• This process makes the fraction easier to work with.

Remember, simplifying rational expressions helps solve more complex problems quicker and cleaner.

factoring

Factoring is breaking down a complex expression into simpler parts that, when multiplied together, give back the original expression.
In the solution, we factored the numerator \(3a + 3\) as \(3(a + 1)\).
Here are some key points to remember about factoring:

• Always search for the greatest common factor (GCF) first. This is the largest number or expression that divides evenly into all terms.
• In \(3a + 3\), the GCF is 3, since both terms are divisible by 3.
• Rewriting \(3a + 3\) as \(3(a + 1)\) shows the GCF outside the parentheses and the simplified expression inside.
• Factoring makes expressions easier to manage and helps in simplifying rational expressions.

The result of factoring plays a critical role in the process of simplifying rational expressions.

reducing fractions

When we simplify fractions, we are essentially reducing them. Reducing fractions means making the numerator and denominator as small as possible while keeping the value of the fraction the same.
Here's how reducing worked in our example: \(\frac{3(a + 1)}{3a}\).
We canceled the common factor (3) in both the numerator and denominator: \frac{3(a + 1)}{3a} = \frac{a + 1}{a}\.
Some tips for reducing fractions:

• Identify common factors in both the numerator and denominator.
• Divide both parts by their common factor.
• The fraction should be in its simplest form with no common factors left.

Reducing fractions is crucial because it simplifies the expression and makes calculations easier.
Always reduce fractions fully to make further steps in solving equations more manageable.