Problem 41 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{\frac{5}{6}}{3+\frac{2}{3}}$$

Step-by-Step Solution

Answer

\(\frac{5}{22}\)

Step 1: Simplify the Denominator Part of the Complex Fraction

We start by simplifying the denominator of the complex fraction, which is \(3 + \frac{2}{3}\). Convert \(3\) into a fraction with a denominator of \(3\), which is \(\frac{9}{3}\). Now the expression is \(\frac{9}{3} + \frac{2}{3}\). Combine these fractions: \[\frac{9}{3} + \frac{2}{3} = \frac{11}{3}\]. Thus, the denominator is simplified to \(\frac{11}{3}\).

Step 2: Simplify the Entire Complex Fraction

Now we have \(\frac{\frac{5}{6}}{\frac{11}{3}}\). To simplify this, we multiply the numerator by the reciprocal of the denominator. Thus, we compute: \[\frac{5}{6} \times \frac{3}{11}\].

Step 3: Perform the Multiplication

Multiply the fractions: \[\frac{5 \cdot 3}{6 \cdot 11} = \frac{15}{66}\].

Step 4: Simplify the Resulting Fraction

Reduce \(\frac{15}{66}\). Both 15 and 66 can be divided by their greatest common divisor, which is 3. Thus, \[\frac{15}{66} = \frac{5}{22}\].

Step 5: Conclusion

The simplified form of the complex fraction \(\frac{\frac{5}{6}}{3+\frac{2}{3}}\) is \(\frac{5}{22}\).

Key Concepts

Simplification of Fractions

Simplifying fractions is crucial when tackling complex fractions. A complex fraction is simply a fraction where the numerator, the denominator, or both are also fractions. The goal is to simplify these expressions to make computations straightforward. For instance, taking a fraction like \( \frac{5}{6} \) over the more complicated \( 3 + \frac{2}{3} \), you start by breaking things down. To simplify the denominator, you first convert whole numbers into fractions with a uniform denominator. This makes addition easier.

• Convert whole numbers into fractions.
• Add fractions by finding a common denominator.
• Simplify any resulting fractions.

By adding the fractions in the denominator here, an expression like \( 3 + \frac{2}{3} \) was converted to \( \frac{11}{3} \), setting the stage for further simplification.

Reciprocal Operation

The reciprocal operation is a vital tool for simplifying complex fractions. This technique involves flipping the numerator and the denominator of a fraction. For instance, the reciprocal of \( \frac{11}{3} \) is \( \frac{3}{11} \). When simplifying a complex fraction, once you have a single fraction as a denominator, you multiply the numerator by the reciprocal of this result. It's like turning division into multiplication, which is often simpler to handle.

• Find the reciprocal by swapping numerator and denominator.
• Multiply the initial fraction by this reciprocal.
• This operation translates the complex fraction into a standard fraction allowing further simplification.

This maneuver precisely transforms \( \frac{\frac{5}{6}}{\frac{11}{3}} \) into a multiplication problem: \( \frac{5}{6} \times \frac{3}{11} \). Therefore, using reciprocal operations is a streamlined method to overcome complex fraction hurdles.

Greatest Common Divisor

A well-known approach to simplify simplified fractions even further involves using the Greatest Common Divisor (GCD). The GCD of two numbers is the largest number that divides both without leaving a remainder. Simplifying fractions means reducing them to their simplest form by dividing the numerator and the denominator by their GCD.

• Calculate the GCD of the numerator and the denominator.
• Divide both the numerator and denominator by this GCD.
• This process ensures you have the simplest form of the given fraction.

Upon multiplying and reaching \( \frac{15}{66} \), apply the GCD method. The GCD of 15 and 66 is 3, leading to the final step where dividing both by 3 results in the simplified fraction, \( \frac{5}{22} \). Hence, utilizing the GCD is a reliable strategy to ensure your fractions are in their most reduced form.