Problem 8 - Algebra for College Students

Question

Simplify each complex fraction. Use either method. $$\frac{\frac{2}{5}-\frac{x}{9}-\frac{1}{3}}{\frac{1}{3}+\frac{x}{5}+\frac{2}{15}}$$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \(\frac{3 - 5x}{21 + 9x}\).

Step 1: Simplify the Numerator

Combine the terms in the numerator: \(\frac{2}{5} - \frac{x}{9} - \frac{1}{3}\). Find the common denominator, which is 45.

Step 1a: Express each term with the common denominator

Convert each fraction as follows: \(\frac{2}{5} = \frac{18}{45}\), \(\frac{x}{9} = \frac{5x}{45}\), and \(\frac{1}{3} = \frac{15}{45}\).

Step 1b: Combine the fractions

Combine: \(\frac{18}{45} - \frac{5x}{45} - \frac{15}{45} = \frac{18 - 5x - 15}{45} = \frac{3 - 5x}{45}\).

Step 2: Simplify the Denominator

Combine the terms in the denominator: \(\frac{1}{3} + \frac{x}{5} + \frac{2}{15}\). Find the common denominator, which is 15.

Step 2a: Express each term with the common denominator

Convert each fraction as follows: \(\frac{1}{3} = \frac{5}{15}\), \(\frac{x}{5} = \frac{3x}{15}\), and \(\frac{2}{15}\) remains the same.

Step 2b: Combine the fractions

Combine: \(\frac{5}{15} + \frac{3x}{15} + \frac{2}{15} = \frac{5 + 3x + 2}{15} = \frac{7 + 3x}{15}\).

Step 3: Combine Numerator and Denominator

Form the complex fraction \(\frac{\frac{3 - 5x}{45}}{\frac{7 + 3x}{15}}\). Simplify by multiplying by the reciprocal of the denominator: \(\frac{3 - 5x}{45} \times \frac{15}{7 + 3x}\).

Step 4: Simplify the Expression

Simplify the multiplication: \(\frac{(3 - 5x) \times 15}{45 \times (7 + 3x)} = \frac{3 - 5x}{3(7 + 3x)} = \frac{3 - 5x}{21 + 9x}\).

Key Concepts

Common Denominator

When working with fractions, finding a common denominator is essential. It allows us to combine fractions by expressing them with the same bottom number. This process simplifies addition and subtraction. In our given complex fraction, we first simplify the numerator: \( \frac{2}{5} - \frac{x}{9} - \frac{1}{3} \). The common denominator for 5, 9, and 3 is 45.
We convert each fraction: \( \frac{2}{5} = \frac{18}{45}, \frac{x}{9} = \frac{5x}{45}, \frac{1}{3} = \frac{15}{45} \).
Combining them, we get: \( \frac{18 - 5x - 15}{45} = \frac{3 - 5x}{45} \). For the denominator of the complex fraction, we need to combine \( \frac{1}{3} + \frac{x}{5} + \frac{2}{15} \). The common denominator here is 15. Convert each term: \( \frac{1}{3} = \frac{5}{15}, \frac{x}{5} = \frac{3x}{15}, \frac{2}{15} = \frac{2}{15} \). Combining them: \( \frac{5 + 3x + 2}{15} = \frac{7 + 3x}{15} \).

Fraction Operations

Performing operations on fractions involves a few key steps. Once fractions have a common denominator, you can easily add or subtract them.\
For addition: sum the numerators and keep the common denominator.
For subtraction: subtract the numerators while maintaining the same denominator.\
In our example, we add and subtract fractions like this: \( \frac{18}{45} - \frac{5x}{45} - \frac{15}{45} = \frac{3 - 5x}{45} \).\
For the denominator, the operation is: \( \frac{5}{15} + \frac{3x}{15} + \frac{2}{15} = \frac{7 + 3x}{15} \). Combining these simplified results is another fraction operation, but here it forms a complex fraction. To deal with this, we utilize the reciprocal of the denominator.

Algebraic Simplification

Simplifying algebraic expressions often involves gathering like terms and reducing fractions. After combining fractions, you might end up with an expression featuring both numbers and variables.\
For our problem, we look at the fraction \( \frac{\frac{3 - 5x}{45}}{\frac{7 + 3x}{15}} \).\
To simplify, multiply by the reciprocal of the denominator: \( \frac{3 - 5x}{45} \times \frac{15}{7 + 3x} \).
The expression now becomes \( \frac{(3 - 5x) \times 15}{45 \times (7 + 3x)} = \frac{3 - 5x}{3(7 + 3x)} = \frac{3 - 5x}{21 + 9x} \). Algebraically, we've moved from a nested fraction to a simpler form.

Reciprocal

The reciprocal of a number or fraction is what you multiply it by to get 1. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). Using the reciprocal is crucial for dividing fractions.\
In our problem, we had the complex fraction \( \frac{\frac{3 - 5x}{45}}{\frac{7 + 3x}{15}} \).
Here, the numerator \( \frac{3 - 5x}{45} \) stays, and we multiply by the reciprocal of the denominator \( \frac{7 + 3x}{15} \). So, it becomes: \( \frac{3 - 5x}{45} \times \frac{15}{7 + 3x} \).
This step turns a complicated division into straightforward multiplication, greatly easing simplification. The final simplified answer is \( \frac{3 - 5x}{21 + 9x} \).