Problem 31 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{\frac{9}{20}-\frac{1}{10}}{\frac{1}{10}+\frac{9}{20}}$$

Step-by-Step Solution

Answer

The simplified complex fraction is \( \frac{7}{11} \).

Step 1: Analyze the Complex Fraction

The given expression is \( \frac{\frac{9}{20}-\frac{1}{10}}{\frac{1}{10}+\frac{9}{20}} \). It is a complex fraction with a fraction in both the numerator and the denominator. Our goal is to simplify it.

Step 2: Find a Common Denominator for the Numerator

The fractions in the numerator are \( \frac{9}{20} \) and \( \frac{1}{10} \). The common denominator is 20. Convert \( \frac{1}{10} \) to have 20 as a denominator: \( \frac{1}{10} = \frac{2}{20} \). Now the numerator is \( \frac{9}{20} - \frac{2}{20} = \frac{7}{20} \).

Step 3: Find a Common Denominator for the Denominator

Similarly, for the denominator with fractions \( \frac{1}{10} \) and \( \frac{9}{20} \), the common denominator is 20. Convert \( \frac{1}{10} \) to \( \frac{2}{20} \), then add to get \( \frac{9}{20} + \frac{2}{20} = \frac{11}{20} \).

Step 4: Simplify the Complex Fraction

Now we have the simplified expression \( \frac{\frac{7}{20}}{\frac{11}{20}} \). To simplify, invert the denominator and multiply: \( \frac{7}{20} \times \frac{20}{11} \).

Step 5: Perform the Multiplication

Cancelling the 20s, we are left with \( \frac{7}{11} \). Thus, the simplified form of the complex fraction is \( \frac{7}{11} \).

Key Concepts

Common Denominator

A common denominator is essential when working with fractions that need to be combined, such as in addition or subtraction. It refers to a shared multiple of the denominators of two or more fractions. Finding a common denominator ensures that the fractions have the same size "parts," allowing them to be easily added or subtracted. In our example, to simplify the complex fraction

• The numerators and denominators involve fractions like \( \frac{9}{20} \) and \( \frac{1}{10} \).
• The first step is to determine a common denominator for each set of these fractions.
• For both, we see that 20 is a convenient common denominator for 10 and 20.

Without a common denominator, aligning these fractions for arithmetic operations would be difficult. When you convert each fraction to have the same denominator, you essentially create a shared playing field, making it easy to mix or compare them. Notice how the conversion step is crucial as it paves the way for straightforward addition or subtraction.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form, in which the numerator and denominator have no common factors other than 1. This means seeing if the numbers can be evenly divided by the same number. In the process of simplifying complex fractions, simplifying each component fraction is crucial.

• After ensuring a common denominator for the fractions in the numerator and denominator, calculate the resulting fraction.
• For instance, with \( \frac{9}{20} - \frac{2}{20} \), you obtain \( \frac{7}{20} \).
• Similarly, adding fractions in the denominator like \( \frac{9}{20} + \frac{2}{20} \) gives \( \frac{11}{20} \).

Once we have the simple fractions, the goal is to simplify them by removing the common denominators via inversions and multiplication. This can lead us directly to a reduced form, like our result of \( \frac{7}{11} \), ensuring no further simplification is needed.

Fraction Operations

Fraction operations involve the basic arithmetic operations — addition, subtraction, multiplication, and division — as applied specifically to fractions. In simplifying a complex fraction like

• We first deal with addition and subtraction by finding common denominators and then perform the operations.
• To handle division in complex fractions, invert the second fraction and multiply.
• This means flipping the fraction in the denominator and then multiplying across the fractions.

For the complex fraction \( \frac{\frac{7}{20}}{\frac{11}{20}} \), inversion allows multiplication resulting in \( \frac{7}{20} \times \frac{20}{11} \). After canceling the 20s, multiplication yields \( \frac{7}{11} \). This integration of fraction operations ensures that each component is properly managed to produce an accurate, simplified result.