Problem 40 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified complex fraction is \(\frac{66}{13}\).

Step 1: Simplify the Numerator

The given numerator is \(3 + \frac{5}{2}\). First, rewrite 3 as a fraction with the same denominator as \(\frac{5}{2}\). Thus, 3 becomes \(\frac{6}{2}\). Now, combine these fractions: \(\frac{6}{2} + \frac{5}{2} = \frac{11}{2}\).

Step 2: Simplify the Denominator

The given denominator is \(\frac{5}{6} + \frac{1}{4}\). To add these fractions, find a common denominator. The least common multiple of 6 and 4 is 12. Convert each fraction: \(\frac{5}{6} = \frac{10}{12}\) and \(\frac{1}{4} = \frac{3}{12}\). Add them to get \(\frac{10}{12} + \frac{3}{12} = \frac{13}{12}\).

Step 3: Simplify the Complex Fraction

Now the complex fraction is \(\frac{\frac{11}{2}}{\frac{13}{12}}\). Simplify by multiplying the numerator by the reciprocal of the denominator: \(\frac{11}{2} \times \frac{12}{13}\). Multiply straight across to get \(\frac{132}{26}\).

Step 4: Simplify the Resulting Fraction

To simplify \(\frac{132}{26}\), find the greatest common divisor (GCD) of 132 and 26, which is 2. Divide the numerator and the denominator by their GCD: \(\frac{132 \div 2}{26 \div 2} = \frac{66}{13}\).

Key Concepts

Simplifying Fractions

Understand that a fraction represents part of a whole. A complex fraction is a fraction where either the numerator, denominator, or both are themselves fractions. Simplifying fractions means reducing them to their simplest form. You do this by finding and using the greatest common divisor (GCD) for both numerator and denominator.

To simplify a complex fraction:

• First, get the numerator and denominator fractions to a manageable form.
• In our exercise, the initial numerator is \(3 + \frac{5}{2}\). Convert 3 into a fraction \(\frac{6}{2}\), then add to get \(\frac{11}{2}\).

The denominator \(\frac{5}{6} + \frac{1}{4}\) needs a common denominator to add conveniently. Use the least common multiple (LCM) to convert both fractions, then add to simplify.

After obtaining a single fraction in the numerator and denominator, simplify further by multiplying the numerator by the reciprocal of the denominator. The result is a single fraction ready for simplification using the GCD, resulting in the simplest form.

Least Common Multiple

When adding fractions, they must share the same bottom number, known as the denominator, to be combined directly. The least common multiple (LCM) comes into play here. It's the smallest number that is a multiple of the denominators involved.

In this complex fraction exercise, the denominators of \(\frac{5}{6}\) and \(\frac{1}{4}\) need a common base:

• The denominators are 6 and 4.
• The multiples of 6 are 6, 12, 18, 24, etc.
• The multiples of 4 are 4, 8, 12, 16, etc.
• The smallest number common in both lists is 12, so that's our LCM.

Use this LCM to adjust the fractions so that their denominators match. Once both fractions share a common denominator (12 in our case), they can be directly added, simplifying the denominator part of the complex fraction.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is a crucial tool for simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder.

In this exercise, after multiplying the simplified fractions, you get \(\frac{132}{26}\). We need to reduce this fraction.

• Find the factors of both 132 and 26.
• The factors of 132 include 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132.
• The factors of 26 are 1, 2, 13, and 26.
• The greatest common factor is 2.

Use this GCD to divide both the numerator and denominator: \(\frac{132}{2} = 66\) and \(\frac{26}{2} = 13\). Therefore, the simplified fraction is \(\frac{66}{13}\). This process ensures we have the simplest form of any fraction.