Problem 45 - Algebra for College Students

Question

For Problems 41-64, simplify each complex fraction. $$ \frac{\frac{5}{6 y}}{\frac{10}{3 x y}} $$

Step-by-Step Solution

Answer

\( \frac{x}{4} \)

Step 1: Understand the Problem

The problem is to simplify a complex fraction, which is a fraction within a fraction. The expression is \( \frac{\frac{5}{6y}}{\frac{10}{3xy}} \). Our task is to simplify this to a simpler form.

Step 2: Recognize the Division of Fractions

A complex fraction can be simplified by transforming the division into multiplication. The division \( \frac{a}{b} \div \frac{c}{d} \) is equivalent to \( \frac{a}{b} \times \frac{d}{c} \). Applying this to our fraction, we have \( \frac{5}{6y} \div \frac{10}{3xy} \equiv \frac{5}{6y} \times \frac{3xy}{10} \).

Step 3: Multiply the Fractions

Multiply the two fractions: \( \frac{5}{6y} \times \frac{3xy}{10} = \frac{5 \times 3xy}{6y \times 10} \). This results in \( \frac{15xy}{60y} \).

Step 4: Simplify the Result

To simplify \( \frac{15xy}{60y} \), begin by cancelling common factors in the numerator and denominator. First, cancel \( y \) from both, leaving \( \frac{15x}{60} \). Next, simplify \( \frac{15x}{60} \) by dividing both the numerator and the denominator by 15, the greatest common divisor. This yields \( \frac{x}{4} \).

Key Concepts

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than one. In the complex fraction \( \frac{\frac{5}{6y}}{\frac{10}{3xy}} \), we start by transforming the division into multiplication by the reciprocal, leading to \( \frac{5}{6y} \times \frac{3xy}{10} \).

• To simplify, multiply the numerators together and the denominators together.
• Therefore, \( 5 \times 3xy = 15xy \) for the numerator, and \( 6y \times 10 = 60y \) for the denominator.

Fraction simplification is integral to solving problems efficiently. Cancel common factors between the numerator and denominator to get the simplified form quickly. This process can seem daunting at first, but with practice, it becomes intuitive. Remember that the goal is to make complex fractions much easier to work with.

Multiplying Fractions

Multiplying fractions is straightforward. You multiply the numerators together and then the denominators together. In the instance of our complex fraction,\[\frac{5}{6y} \times \frac{3xy}{10} = \frac{15xy}{60y} \].
Here are the steps to keep in mind:

• Multiply the numerators: This creates the new numerator.
• Multiply the denominators: This forms the new denominator, which simplifies the complexity of the fraction.
• Cancel and simplify: Look out for common terms or factors across numerators and denominators that you can cancel out to simplify the fraction.

This approach makes handling even complex-looking fractions manageable and is a fundamental skill in algebra and calculus. Multiplying fractions like this can be practiced to enhance your math solving skills dramatically.

Greatest Common Divisor

The greatest common divisor (GCD) is essential in simplifying fractions. It is the largest number that evenly divides each of the numbers. In our fraction \(\frac{15xy}{60y}\), in order to simplify it, we need to use the GCD.
Here's how you can simplify effectively:

• Cancel common variables: If both the numerator and the denominator have the same variable terms, like \(y\) in this case, cancel them out first.
• Identify the GCD of the coefficients: For 15 and 60, the GCD is 15. Divide both numerator and denominator by 15.

This gives us the simplified fraction \( \frac{x}{4} \).Understanding and applying the GCD is not only crucial for simplifying fractions but is also widely used in solving equations and other mathematical practices.