Problem 4 - Algebra 2 and Trigonometry

Question

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{\frac{2}{5}}{4}\)

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{1}{10} \), and it is always defined.

Step 1: Identify the Inner Fraction

Observe the given complex rational expression: \( \frac{\frac{2}{5}}{4} \). Identify the inner fraction. In this expression, the inner fraction is \( \frac{2}{5} \).

Step 2: Interpret the Fraction

The given expression can be interpreted as dividing the inner fraction by 4. This is equivalent to multiplying the inner fraction by the reciprocal of 4. The reciprocal of 4 is \( \frac{1}{4} \).

Step 3: Perform the Multiplication

Multiply \( \frac{2}{5} \) by \( \frac{1}{4} \): \[\frac{2}{5} \times \frac{1}{4} = \frac{2 \times 1}{5 \times 4} = \frac{2}{20}\]

Step 4: Simplify the Resulting Fraction

Simplify \( \frac{2}{20} \). Notice that both the numerator and the denominator can be divided by 2 to simplify the expression:\[\frac{2}{20} = \frac{1}{10}\]

Step 5: Identify Undefined Values

In this problem, the only fractions involved are \( \frac{2}{5} \) and \( \frac{1}{4} \). The values for which the fractions are not defined would be when the denominators are zero. Since the denominators 5 and 4 are never zero, the expression is always defined.

Key Concepts

Fraction Simplification

Fraction simplification is an essential aspect of working with fractions as it allows for easier interpretation and manipulation of these numbers. When simplifying a fraction, the goal is to reduce it to its simplest form. This is achieved by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number. In the expression \( \frac{2}{20} \), we simplified by dividing both the numerator and the denominator by 2, resulting in \( \frac{1}{10} \). Simplifying fractions helps in achieving clear and reduced forms without changing the fraction's value. Simplification aligns with the principle that a fraction represents a division; the fewer the factors, the cleaner the expression.

Undefined Values in Fractions

Understanding undefined values in fractions is crucial to solving fraction problems correctly. A fraction is undefined when its denominator is zero because division by zero is mathematically impossible.
In any expression involving fractions, it's important to check what values of the variable could make the denominator zero.
This becomes especially relevant when variables are involved. In this case, since the denominators of both \( \frac{2}{5} \) and \( \frac{1}{4} \) are constants (5 and 4, respectively), these fractions are always defined. Remember, check the denominators first to prevent any undefined scenarios.

Multiplication of Fractions

The multiplication of fractions is a straightforward process that requires multiplying the numerators together and the denominators together. When you multiply two fractions, such as \( \frac{2}{5} \) and \( \frac{1}{4} \), the result is a fraction formed by the product of the two numerators over the product of the two denominators.

• Multiply the numerators: \(2 \times 1 = 2\)
• Multiply the denominators: \(5 \times 4 = 20\)

This results in \( \frac{2}{20} \), which can often be further simplified.
Understanding this fundamental operation aids in a variety of mathematical topics and real-life applications involving ratios, rates, and proportional reasoning.