Problem 25 - Algebra 2

Question

Simplify each complex fraction. \(\frac{\frac{1}{3}}{\frac{3}{b}}\)

Step-by-Step Solution

Answer

The simplified form of the complex fraction \(\frac{\frac{1}{3}}{\frac{3}{b}}\) is \(\frac{b}{9}\).

Step 1: Identify the complex fraction and the reciprocal of the denominator

The complex fraction presented is \(\frac{\frac{1}{3}}{\frac{3}{b}}\). The reciprocal of the denominator, \(\frac{3}{b}\), is the fraction flipped, which would give \(\frac{b}{3}\).

Step 2: Multiply the numerator by the reciprocal of the denominator

After identifying the complex fraction and the reciprocal of the denominator, the step is to multiply the numerator of the complex fraction (\(\frac{1}{3}\)) by the reciprocal of the denominator (\(\frac{b}{3}\)). This process essentially turns the division operation into a multiplication operation, which is easier to calculate. Doing so will result in \(\frac{1}{3} * \frac{b}{3}\), which further simplifies to \(\frac{b}{9}\).

Key Concepts

Simplifying Fractional Expressions

Simplifying fractional expressions might seem tricky at first glance, but with a structured approach, it becomes much more manageable. A fractional expression is essentially a mathematical expression that includes fractions. When "simplifying" a fractional expression, the goal is to rewrite it in the simplest form without changing its value. This often involves reducing the numerator and denominator to their smallest possible integers or expressions.

In the case of complex fractions, which are fractions where the numerator, the denominator, or both are also fractions, simplification involves a few additional steps. Here's how it generally works:

• Identify the numerator and the denominator of the complex fraction separately.
• Locate the reciprocal of the denominator fraction.
• Change the division of fractions problem into a multiplication problem by using this reciprocal.
• Perform the multiplication to arrive at a simpler expression.
• Finally, reduce the final fraction to its simplest form by factoring out common factors if possible.

This process turns what might appear to be a complicated problem into an understandable and more straightforward computation.

Reciprocal

The concept of a reciprocal is central when dealing with fractions, especially in operations involving complex fractions. A reciprocal of a fraction is simply that fraction flipped upside down.

For example, if you have a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). Finding the reciprocal is straightforward, yet powerful, because it allows one to convert division problems into multiplication problems. This conversion greatly simplifies the task:

• Identify the fraction you want to find the reciprocal of.
• Swap the numerator and denominator positions.
• The result is the reciprocal of your original fraction.

For instance, in the exercise we reviewed, the reciprocal of \( \frac{3}{b} \) is \( \frac{b}{3} \). Using a reciprocal is essential in operations like division, as it helps to scale the numbers without changing the value of the expressions being operated on, ultimately simplifying the process.

Multiplying Fractions

Multiplying fractions is one of the simpler arithmetic operations involving fractions but essential, especially when dealing with complex fractions. Unlike adding or subtracting fractions, multiplication does not require common denominators.

To multiply fractions:

• Multiply the numerators of the fractions together to get the new numerator.
• Multiply the denominators of the fractions together to get the new denominator.
• Simplify the resulting fraction if possible by reducing it to its lowest terms.

For example, if you multiply \( \frac{1}{3} \) by \( \frac{b}{3} \), you multiply the tops: \( 1 \times b = b \); and the bottoms: \( 3 \times 3 = 9 \). This results in \( \frac{b}{9} \). This rule makes multiplying fractions straightforward compared to other operations, and it's particularly useful for simplifying complex fractions once the reciprocal is applied.