Problem 38 - Algebra for College Students

Question

Simplify each complex fraction. $$\frac{\frac{1}{y+2}-\frac{4}{3 y}}{\frac{3}{y}-\frac{2}{y+3}}$$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{-(y+8)(y+3)}{3(y+2)(y+9)} \).

Step 1: Simplify the Numerator

Combine the fractions in the numerator. The given fraction in the numerator is \ \ \( \frac{1}{y+2} - \frac{4}{3y} \). Find a common denominator, which will be \( (y+2)(3y) \). Rewrite each fraction: \ \ \( \frac{1}{y+2} = \frac{3y}{(y+2)(3y)} \) \ \ \( \frac{4}{3y} = \frac{4(y+2)}{(y+2)(3y)} \). \ \ Therefore, the numerator becomes: \ \ \( \frac{3y - 4(y+2)}{(y+2)(3y)} = \frac{3y - 4y - 8}{(y+2)(3y)} = \frac{-y - 8}{3y(y+2)} \).

Step 2: Simplify the Denominator

Combine the fractions in the denominator. The given fraction in the denominator is \ \ \( \frac{3}{y} - \frac{2}{y+3} \). Find a common denominator, which will be \( y(y+3) \). Rewrite each fraction: \ \ \( \frac{3}{y} = \frac{3(y+3)}{y(y+3)} \) \ \ \( \frac{2}{y+3} = \frac{2y}{y(y+3)} \). \ \ Therefore, the denominator becomes: \ \ \( \frac{\frac{3(y+3) - 2y}{y(y+3)}} = \frac{3y + 9 - 2y}{y(y+3)} = \frac{y + 9}{y(y+3)} \).

Step 3: Divide the Fractions

The original complex fraction can now be rewritten as: \ \ \( \frac{\frac{-y - 8}{3y(y+2)}}{\frac{y + 9}{y(y+3)}} \). \ \ To divide by a fraction, multiply by its reciprocal: \ \ \( \frac{-y - 8}{3y(y+2)} \times \frac{y(y+3)}{y + 9} = \frac{(-y - 8) \times y(y+3)}{3y(y+2) \times (y + 9)} \).

Step 4: Simplify the Expression

Multiply the numerators and denominators together: \ \ \( \frac{(-y - 8) \times y(y+3)}{3y(y+2) \times (y + 9)} = \frac{-(y+8) \times y \times (y+3)}{3y \times (y+2) \times (y+9)} \). \ \ Divide out common factors: \ \ The \( y \) in the numerator and the denominator cancel out, leaving: \ \ \( \frac{-(y+8) \times (y+3)}{3 \times (y+2) \times (y+9)} \). \ \ Therefore, the simplified fraction is: \( \frac{-(y+8)(y+3)}{3(y+2)(y+9)} \).

Key Concepts

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. These expressions can include variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division.
An example of an algebraic fraction is \(\frac{3x + 2}{x^2 - 1}\). Simplifying these fractions involves reducing the algebraic expressions to their simplest form.
Just like numeric fractions, algebraic fractions can be simplified by factoring and cancelling out common factors from the numerator and the denominator.

Numerator and Denominator

In any fraction, the top part is called the numerator, and the bottom part is called the denominator. For instance, in the fraction \(\frac{a}{b}\), 'a' is the numerator, and 'b' is the denominator.
When dealing with algebraic fractions, it's essential to understand the roles of the numerator and the denominator as they determine the overall value of the fraction. To simplify complex fractions, we often need to manipulate both the numerator and the denominator.
For example, in the given problem, we have to find common denominators for both the numerator and the denominator to combine them into simpler fractions before further simplifying.

Reciprocals

Reciprocals are simply the 'flipped' version of a fraction. If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). Reciprocals are crucial when dividing fractions, including algebraic ones.
For example, to divide by a fraction such as \(\frac{3}{4}\), you multiply by its reciprocal \(\frac{4}{3}\).
In the given problem, we use the reciprocal when dividing the complex fraction. Instead of dividing by the second algebraic fraction in the denominator, we multiply by its reciprocal. This makes the complex fraction easier to handle and simplifies the overall expression.

Common Denominators

Finding common denominators is essential when adding or subtracting fractions. A common denominator allows us to combine fractions into a single fraction.
For example, if we have \(\frac{1}{y + 2}\) and \(\frac{4}{3y}\), we need a common denominator to combine these into one fraction. The common denominator would be \((y + 2)(3y)\).
Similarly, for the denominator fractions in the original problem, we find a common denominator like \((y)(y + 3)\) to combine \(\frac{3}{y}\) and \(\frac{2}{y+3}\). This allows us to simplify the complex algebraic fraction step by step, ultimately making the original fraction simpler to handle.