Problem 35 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{2+\frac{5}{6}}{5-\frac{1}{3}}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{17}{28}\).

Step 1: Simplify the Numerator

The numerator of the complex fraction is \(2 + \frac{5}{6}\). To combine these into a single fraction, express \(2\) as a fraction with a denominator of 6, which becomes \(\frac{12}{6}\). Adding these two fractions, we get \(\frac{12}{6} + \frac{5}{6} = \frac{17}{6}\).

Step 2: Simplify the Denominator

The denominator of the complex fraction is \(5 - \frac{1}{3}\). Express \(5\) as a fraction with a denominator of 3, which is \(\frac{15}{3}\). Subtract \(\frac{1}{3}\) from \(\frac{15}{3}\), resulting in \(\frac{15}{3} - \frac{1}{3} = \frac{14}{3}\).

Step 3: Set Up Division of Fractions

The simplified complex fraction is now \(\frac{\frac{17}{6}}{\frac{14}{3}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so we rewrite this as \(\frac{17}{6} \times \frac{3}{14}\).

Step 4: Multiply the Fractions

To multiply \(\frac{17}{6}\) and \(\frac{3}{14}\), multiply the numerators (17 and 3) and the denominators (6 and 14): \[ \frac{17 \times 3}{6 \times 14} = \frac{51}{84} \].

Step 5: Simplify the Resulting Fraction

Find the greatest common factor of 51 and 84, which is 3. Divide both the numerator and denominator by 3: \[ \frac{51 \div 3}{84 \div 3} = \frac{17}{28} \]. This is the simplest form of the fraction.

Key Concepts

Simplifying Fractions

Understanding how to simplify fractions is essential when working with complex fractions. When we simplify a fraction, we are reshaping it into its simplest form, meaning the numerator and the denominator are as small as they can possibly be while still maintaining the same value. To do this, we look for a common factor in both the numerator and the denominator. We perform this by:

• Identifying the greatest common factor (GCF) of the numerator and the denominator. This is the largest number that can divide both the numerator and the denominator without any remainder.
• Dividing both the numerator and the denominator by their GCF. This process reduces the numbers but preserves the actual value of the fraction.

In the solution above, after calculating the product of the fractions, we found that the fraction \( \frac{51}{84} \) can be simplified because both 51 and 84 share a common factor, which is 3. After dividing them by the GCF, we reach \( \frac{17}{28} \), the simplest form.

Numerators and Denominators

In every fraction, the top number is called the numerator, and the bottom number is the denominator. Each has a specific role:

• Numerator: Indicates how many parts we are considering or have.
• Denominator: Represents the total number of equal parts in a whole.

When working with complex fractions like \( \frac{2+\frac{5}{6}}{5-\frac{1}{3}} \), it's crucial first to simplify both the numerator and the denominator independently. This involves:

• For the numerator \(2 + \frac{5}{6}\): Converting whole numbers to have the same denominator as the fraction to combine them easily.
• For the denominator \(5 - \frac{1}{3}\): A similar conversion is used to allow subtraction by expressing the whole number with a denominator of 3.

This eases the manipulation required to simplify complex fractions, allowing us to handle them as regular fractions.

Multiplication of Fractions

The multiplication of fractions is a straightforward process that involves multiplying the numerators together and the denominators together. This concept is particularly useful when dealing with complex fractions, as division by a fraction is converted into multiplication by its reciprocal.

Here's what you need to remember:

• Reciprocal: The reciprocal of a fraction is created by swapping its numerator and denominator. This means for \(\frac{1}{3}\), the reciprocal is \(\frac{3}{1}\).
• When dividing fractions, as in the example \(\frac{\frac{17}{6}}{\frac{14}{3}}\), we multiply \(\frac{17}{6}\) by the reciprocal of \(\frac{14}{3}\).
• This results in: \(\frac{17}{6} \times \frac{3}{14}\), which simplifies to \(\frac{51}{84}\).

Always ensure to simplify the resulting fraction whenever possible, achieving the simplest expression of your solution. This keeps calculations clean and manageable.