Problem 62 - Algebra for College Students

Question

Simplify each complex fraction. $$ \frac{a}{\frac{1}{a}+4}+1 $$

Step-by-Step Solution

Answer

\( \frac{a^2 + 4a + 1}{1 + 4a} \)

Step 1: Identify the Fraction Structure

We are given the complex fraction \( \frac{a}{\frac{1}{a} + 4} + 1 \). Notice that the denominator itself is a fraction, \( \frac{1}{a} \), added with 4.

Step 2: Simplify the Denominator

Focus on the denominator \( \frac{1}{a} + 4 \). To add these, find a common denominator. In this case, that would be \(a\). Rewrite the expression as \( \frac{1}{a} + \frac{4a}{a} = \frac{1 + 4a}{a} \).

Step 3: Simplify the Complex Fraction

Now substitute the simplified denominator back into the main fraction. The expression becomes \( \frac{a}{\frac{1 + 4a}{a}} + 1 \). Simplify \( \frac{a}{\frac{1 + 4a}{a}} \) by multiplying \( a \) by the reciprocal of \( \frac{1 + 4a}{a} \), which is \( \frac{a}{1 + 4a} \), resulting in \( \frac{a \cdot a}{1 + 4a} = \frac{a^2}{1 + 4a} \).

Step 4: Final Expression

Include the 1 from the original expression: \( \frac{a^2}{1 + 4a} + 1 \). To combine, convert 1 into a fraction with a denominator of \(1 + 4a\): \( \frac{a^2}{1 + 4a} + \frac{1 + 4a}{1 + 4a} \).

Step 5: Combine the Fractions

Add the fractions: \( \frac{a^2 + (1 + 4a)}{1 + 4a} = \frac{a^2 + 1 + 4a}{1 + 4a} \).

Step 6: Simplify the Expression

Reorder and combine like terms in the numerator: \( \frac{a^2 + 4a + 1}{1 + 4a} \). The fraction is now in its simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This process makes calculations and comparisons easier.

• A simple fraction is a ratio where both numerator and denominator are whole numbers.
• A complex fraction, however, can have a fraction as its numerator, denominator, or both.

To simplify any fraction, our goal is to find common factors in the numerator and denominator, and then divide both by these common factors. When dealing with complex fractions, it often helps to simplify any fractional parts into simpler whole number equivalents.
For instance, when tackling the complex fraction \( \frac{a}{\frac{1}{a} + 4} + 1 \), we first focused on simplifying the denominator \( \frac{1}{a} + 4 \). By converting all terms to share a common denominator, calculations become straightforward.
This approach minimizes confusion and reduces the fraction to an easier form to work with.

Common Denominator

The concept of a common denominator is fundamental when working with fractions, especially for addition and subtraction. It occurs when different fractions have the same value in the denominator, allowing the combination of the numerators directly.

• Finding a common denominator is essential when adding or subtracting fractions.
• This ensures that the fractions are comparable on the same scale, much like comparing apples to apples.

In the exercise, while simplifying \( \frac{1}{a} + 4 \), the key was to establish a common denominator between "1" and the term "4". We recognized that "4" could be rewritten as \( \frac{4a}{a} \), providing the common denominator \( a \).
This method allowed for a smooth addition process, simplifying complex fractions effortlessly.
By handling fractions in this manner, manipulation and solving become much more manageable.

Algebraic Expressions

Algebraic expressions form the building blocks of algebra. They include numbers, variables, and arithmetic operations. Understanding how to simplify and manipulate these expressions is crucial.

• Variables represent unknown values and can be manipulated like numbers.
• Expressions can have one or more terms combined with operators such as + or -.

When dealing with a complex fraction like \( \frac{a}{\frac{1}{a} + 4} + 1 \), it's essential to be adept at managing algebraic expressions. Simplifying such expressions involves understanding terms like \( a^2 \) or \( 4a \) and basic operations like addition and multiplication.
In our solution, we carefully combined terms to simplify the expression, combining similar terms such as \( a^2 \) and \( 4a \) into a streamlined expression. Reordering and combining like terms in the numerator gave us the final simplified version of the fraction.
Mastery over algebraic expressions ensures accurate and efficient resolving of even the most complex algebraic fractions.