Problem 35 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 3 or \(5 .\) $$ \frac{\frac{3 y}{x}-y}{y-\frac{y}{x}} $$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{3-x}{x-1}\).

Step 1: Identify the Complex Fraction

The given complex fraction is \(\frac{\frac{3y}{x} - y}{y - \frac{y}{x}}\). This is a fraction where both the numerator and the denominator contain fractions themselves.

Step 2: Find a Common Denominator

In both the numerator \(\left(\frac{3y}{x} - y\right)\) and the denominator \(\left(y - \frac{y}{x}\right)\), we have terms involving fractions. The common denominator for both is \(x\).

Step 3: Rewrite the Numerator with Common Denominator

Re-express \(y\) as \(\frac{xy}{x}\) so that both terms in the numerator have \(x\) as a denominator: \(\frac{3y}{x} - \frac{xy}{x}\). This becomes \(\frac{3y - xy}{x}\).

Step 4: Rewrite the Denominator with Common Denominator

Similarly, rewrite \(y\) in the denominator as \(\frac{xy}{x}\): \(\frac{xy}{x} - \frac{y}{x}\). This simplifies to \(\frac{xy - y}{x}\).

Step 5: Simplify the Complex Fraction

The complex fraction \(\frac{\frac{3y-xy}{x}}{\frac{xy-y}{x}}\) can be simplified by multiplying the numerator by the reciprocal of the denominator: \(\frac{3y-xy}{x} \times \frac{x}{xy-y}\). The \(x\) terms cancel out, leaving \(\frac{3y-xy}{xy-y}\).

Step 6: Factor and Simplify If Possible

Factor out \(y\) from the numerator and denominator: \(\frac{y(3-x)}{y(x-1)}\). Cancel \(y\) from the top and bottom to get \(\frac{3-x}{x-1}\).

Step 7: Simplified Fraction

The simplified form of the original complex fraction is \(\frac{3-x}{x-1}\).

Key Concepts

Algebraic Simplification

Algebraic simplification is a fundamental process in algebra that involves reducing expressions to their simplest form. When working with complex fractions, simplification is crucial to make the expressions more manageable and easier to understand.
Here's a step-by-step guide on how to simplify an algebraic expression:

• Identify the terms that can be combined or reduced. Look for like terms or factors that are common throughout the expression.
• Use basic operations, such as addition, subtraction, multiplication, or division, to combine like terms.
• Factor expressions whenever possible. Cancel common factors from the numerator and denominator if applicable.
• Check to ensure all parts of the expression are simplified and that there are no further reductions possible.

For example, in the complex fraction \(\frac{\frac{3y}{x} - y}{y - \frac{y}{x}}\), the simplification process included rewriting terms with a common denominator and factoring out common elements, which ultimately led to the simplified expression \(\frac{3-x}{x-1}\).

Fraction Operations

Fraction operations involve performing mathematical operations such as addition, subtraction, multiplication, and division with fractions. When dealing with complex fractions, understanding these operations is essential to simplify the expressions.
In the context of complex fractions:

• Recognize that a complex fraction is essentially a fraction where both the numerator and the denominator are fractions themselves.
• To simplify, convert the division of fractions into multiplication by the reciprocal. This means if you have \(\frac{a}{b} \div \frac{c}{d}\), it becomes \(\frac{a}{b} \times \frac{d}{c}\).
• Apply this rule to the numerator and denominator separately before simplifying the entire complex fraction.
• Always check your work by ensuring that the fractions were correctly multiplied or divided.

In our example, multiplying by the reciprocal was used to transform the complex fraction \(\frac{\frac{3y-xy}{x}}{\frac{xy-y}{x}}\) into a simpler expression \(\frac{3y-xy}{xy-y}\).

Common Denominators

Finding a common denominator is an essential step when working with fractions, especially complex fractions. This process allows you to rewrite fractions so they can be easily added, subtracted, or compared.
Here's how you can find and use common denominators:

• Identify the denominators of the fractions you are working with. In most cases, you'll be dealing with multiple different denominators.
• Determine the least common denominator (LCD), which is the smallest number that is a multiple of each denominator.
• Rewrite each fraction as an equivalent fraction with the LCD. This involves adjusting the numerators so that the value of the fractions remains the same.
• Once all fractions share a common denominator, it's easy to add or subtract them.

In the problem \(\frac{\frac{3y}{x}-y}{y-\frac{y}{x}}\), rewriting terms using the common denominator \(x\) ensured both parts of the complex fraction could be combined and simplified efficiently, resulting in a simpler equation than the original.