Problem 37 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified complex fraction is \( \frac{23}{16} \).

Step 1: Rewrite the Complex Fraction

The given complex fraction is \( \frac{3 + \frac{5}{6}}{1 + \frac{5}{3}} \). Our first task is to express each part of the fraction as a single fraction. Begin with the numerator: \( 3 + \frac{5}{6} \). To combine these, convert \( 3 \) to a fraction with a denominator of 6: \( \frac{18}{6} + \frac{5}{6} \). The denominator is \( 1 + \frac{5}{3} \). Convert \( 1 \) to a fraction with a denominator of 3: \( \frac{3}{3} + \frac{5}{3} \).

Step 2: Simplify the Numerator

Combine the two fractions in the numerator: \( \frac{18}{6} + \frac{5}{6} = \frac{23}{6} \). Now the numerator is \( \frac{23}{6} \).

Step 3: Simplify the Denominator

Combine the two fractions in the denominator: \( \frac{3}{3} + \frac{5}{3} = \frac{8}{3} \). Now the denominator is \( \frac{8}{3} \).

Step 4: Divide the Fractions

The fraction now looks like \( \frac{\frac{23}{6}}{\frac{8}{3}} \). To divide these fractions, multiply by the reciprocal of the denominator: \( \frac{23}{6} \times \frac{3}{8} \).

Step 5: Perform the Multiplication

Now multiply the fractions: \( \frac{23 \times 3}{6 \times 8} = \frac{69}{48} \).

Step 6: Simplify the Resulting Fraction

Simplify \( \frac{69}{48} \) by finding the greatest common divisor of 69 and 48, which is 3. Divide both the numerator and the denominator by 3: \( \frac{69 \div 3}{48 \div 3} = \frac{23}{16} \).

Step 7: Write the Final Answer in Simplest Form

The simplest form of the original complex fraction \( \frac{3 + \frac{5}{6}}{1 + \frac{5}{3}} \) is \( \frac{23}{16} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is a key skill in arithmetic that helps in reducing fractions to their simplest form. This process involves reducing the numerator and denominator to the smallest possible integers that maintain the same value of the fraction.
When you encounter complex fractions, like the example provided, start by ensuring that individual sections are written as single fractions. This means handling any additions or subtractions by finding a common denominator.

• Convert whole numbers or integers to fractions by giving them the same denominator as the fractional part.
• Add these fractions by combining them over the common denominator.

Once you have the fractions simplified within the numerator and the denominator, proceed to divide or multiply these fractions to resolve the complex fraction entirely. This will involve further simplification.

Reciprocal

The concept of a reciprocal is crucial when working with complex fractions, especially when dividing them. To find the reciprocal of a fraction, simply swap the numerator and the denominator.

For example, the reciprocal of \( \frac{8}{3} \) is \( \frac{3}{8} \). When dividing two fractions, multiply by the reciprocal of the divisor instead of directly dividing. This process changes a division problem into a multiplication problem, making it easier to handle:

• If you need to divide by a fraction, multiply by its reciprocal.
• Perform the multiplication as you usually would with fractions.

In the original exercise, dividing \( \frac{23}{6} \) by \( \frac{8}{3} \) is handled by multiplying \( \frac{23}{6} \times \frac{3}{8} \), simplifying the operations significantly.

Numerator and Denominator

Understanding numerators and denominators is fundamental when working with fractions. The numerator is the top part of a fraction, representing how many parts we are considering, while the denominator is the bottom part, showing the total number of equal parts in the whole. For example, in the fraction \( \frac{23}{16} \), 23 is the numerator, and 16 is the denominator.

When simplifying complex fractions, these elements must be handled carefully.

• Ensure all parts of a fraction are represented with common denominators when adding or subtracting.
• In complex fractions, treat both the numerator and denominator as a single entity when possible to simplify calculations.

Paying attention to numerators and denominators ensures accuracy and clarity in the simplification process, making subsequent arithmetic operations much more manageable.

Greatest Common Divisor

The greatest common divisor (GCD) is instrumental in simplifying fractions to their simplest form. It is the largest integer that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is the key step in reducing a fraction completely.
In the example solution, once the fraction \( \frac{69}{48} \) is obtained, it is simplified to \( \frac{23}{16} \) using the GCD:

• Identify numbers that can divide both the numerator and denominator.
• Use the GCD to divide both, ensuring the fraction remains equivalent in value but is reduced in complexity.

Finding the GCD is useful not only in this specific task but also in general fraction mathematics, helping avoid cumbersome calculations and errors.