Problem 45 - Prealgebra

Question

Simplify each of the following complex fractions. [Examples 5–7] $$\frac{2+1 \frac{2}{3}}{3 \frac{5}{6}-1}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{22}{17}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions. The expression in the numerator is \(2 + 1 \frac{2}{3}\), which simplifies to \(2 + \frac{5}{3}\). For \(1 \frac{2}{3}\), you multiply 1 by 3 and add 2, resulting in \(\frac{5}{3}\). So, \(2\) is \(\frac{6}{3}\), making the entire numerator \(\frac{6}{3} + \frac{5}{3} = \frac{11}{3}\). The denominator is \(3 \frac{5}{6} - 1\). For \(3 \frac{5}{6}\), multiply 3 by 6 and add 5, resulting in \(\frac{23}{6}\). So, the denominator is \(\frac{23}{6} - \frac{6}{6} = \frac{17}{6}\).

Step 2: Write the Division Expression

Now that we have improper fractions, rewrite the complex fraction as a division problem: \(\frac{11}{3} \div \frac{17}{6}\).

Step 3: Multiply by the Reciprocal

To divide by a fraction, you multiply by its reciprocal. Thus, \(\frac{11}{3} \div \frac{17}{6}\) becomes \(\frac{11}{3} \times \frac{6}{17}\).

Step 4: Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{11 \times 6}{3 \times 17}\), simplifying to \(\frac{66}{51}\).

Step 5: Simplify the Resulting Fraction

Finally, simplify \(\frac{66}{51}\). The greatest common divisor of 66 and 51 is 3, so divide both the numerator and the denominator by 3: \(\frac{66 \div 3}{51 \div 3} = \frac{22}{17}\).

Key Concepts

Improper Fractions

When dealing with complex fractions, one common step is converting mixed numbers into improper fractions. A mixed number includes a whole number and a fraction, such as \(1 \frac{2}{3}\). To transform it into an improper fraction:

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• Place the total over the original denominator.

For example, \(1 \frac{2}{3}\) becomes \(\frac{5}{3}\). By handling all components as improper fractions, calculations become much simpler.

Reciprocals in Division

Dividing fractions involves an important concept: reciprocals. A reciprocal is what you multiply a number by to get 1. For fractions, you swap the numerator and denominator. So the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

• When dividing \(\frac{11}{3}\) by \(\frac{17}{6}\), instead multiply by the reciprocal of \(\frac{17}{6}\), which is \(\frac{6}{17}\).
• This changes the division to multiplication: \(\frac{11}{3} \times \frac{6}{17}\).

Reciprocal multiplication is a key step for simplifying complex fractions.

Simplifying Fractions

Once you've resolved an expression involving fractions, you usually need to simplify the result. Simplifying a fraction means converting it to its simplest form, where the numerator and the denominator have no common factors (except 1).

• First, find the greatest common divisor (GCD) of the numerator and the denominator.
• Then divide both the numerator and the denominator by this GCD.

For instance, in the fraction \(\frac{66}{51}\), both numbers can be divided by 3 (the GCD), resulting in the simplest form \(\frac{22}{17}\). Simplifying helps ensure that your fraction is as concise as possible.