Problem 59 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\frac{2 a+2 b}{a} \cdot \frac{1}{2}$$

Step-by-Step Solution

Answer

1 + \(\frac{b}{a}\)

Step 1: Simplify the First Fraction

The given expression is \(\frac{2a + 2b}{a} \times \frac{1}{2}\). Focus on simplifying the numerator of the first fraction. Factor out the common term in the numerator: \(2a + 2b = 2(a + b)\). Thus, the expression becomes \(\frac{2(a + b)}{a} \times \frac{1}{2}\).

Step 2: Simplify the Fraction

Simplify \(\frac{2(a + b)}{a}\) by canceling out the common factors in the numerator and the denominator. \[\frac{2(a + b)}{a} = 2 \times \frac{a + b}{a} = 2 \times (1 + \frac{b}{a}) = 2 \times (1 + \frac{b}{a})\]

Step 3: Multiply by the Second Fraction

Now multiply the simplified fraction by \(\frac{1}{2}\). \(\frac{2(a + b)}{a} \times \frac{1}{2} = (1 + \frac{b}{a}) \times 1 = 1 + \frac{b}{a}\)

Key Concepts

Algebraic Fractions

Algebraic fractions are fractions that contain variables in the numerator, the denominator, or both. Just like regular fractions, algebraic fractions can be added, subtracted, multiplied, and divided. Algebraic fractions appear in many algebraic problems and are essential in simplifying expressions or solving equations. Understanding algebraic fractions involves knowing how to factor polynomials and simplify expressions, as this helps in performing operations correctly.
The fraction in our problem \(\frac{2a + 2b}{a}\) is an algebraic fraction because it includes variables \(a\) and \(b\) in its numerator and \(a\) in its denominator.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form by performing operations and combining like terms. This often involves factoring, canceling out common factors, and reorganizing terms to make the expression more manageable. In our exercise, the initial step involves factoring the numerator of the first fraction. We have \(2a + 2b\). Notice that 2 is a common factor in both terms. Therefore, we can factor this out to get \(2(a + b)\).
This transforms our fraction to \(\frac{2(a + b)}{a}\). After factoring, simplifying continues by canceling out common factors in the numerator and denominator. In this case, the \(2\) and \(a\) terms allow us to simplify \(2 \times \frac{(a + b)}{a}\).
The next steps involve simplifying further to reach our final simplified form.

Multiplication of Fractions

The multiplication of fractions involves multiplying the numerators together and the denominators together. When dealing with algebraic fractions, the same principles apply. After simplifying the expression, we multiply the simplified fraction by the second fraction. In our simplification exercise, the expression \(\frac{2(a + b)}{a} \times \frac{1}{2}\) becomes very manageable once simplified.
First, simplify the first fraction to get \(2 \times (1 + \frac{b}{a})\) then the final simplification step is applying the multiplication rule: \(2 \times (1 + \frac{b}{a}) \times \frac{1}{2} = 1 + \frac{b}{a}\). Notice how multiplying effectively simplified our expression to its simplest form.
Understanding the rules of multiplying fractions, both regular and algebraic, helps solve these problems efficiently and correctly.