Question

Simplify. Leave your answers as improper fractions. $$\frac{x y-\frac{3 x}{a c}}{\frac{a c}{x}+2 c}$$

Short answer

\[\frac{acy - 3}{ac + 2cx}\]

Step-by-step solution

Find a common denominator

In order to simplify the expression, first, find a common denominator for the two fractions in the numerator. The common denominator is going to be 'ac'. Multiply the first term 'xy' by 'ac/ac' to get the common denominator.

Rewrite the expression

Rewrite the expression with a common denominator in the numerator: \[\frac{\frac{acxy}{ac} - \frac{3x}{ac}}{\frac{ac}{x}+2c}\]

Combine terms over the common denominator

Combine the terms in the numerator over the common denominator to get a single fraction in the numerator: \[\frac{acxy - 3x}{ac}\]

Simplify the numerator

Simplify the numerator by factoring out the common term 'x': \[\frac{x(acy - 3)}{ac}\]

Simplify the denominator

Rewrite the terms in the denominator to have a common denominator of 'x': \[\frac{ac}{x} + 2c = \frac{ac + 2cx}{x}\]

Combine the numerator and denominator

Place the simplified numerator over the simplified denominator: \[\frac{x(acy - 3)}{ac(\frac{ac + 2cx}{x})}\]

Simplify the complex fraction

Simplify the complex fraction by multiplying the top and bottom by 'x': \[\frac{x(acy - 3)}{ac} \times \frac{x}{ac + 2cx}\]

Cancel the x terms

Cancel the 'x' terms in the numerator and the 'x' in the denominator of the complex fraction: \[\frac{acy - 3}{ac + 2cx}\]

Final answer

The expression is now simplified to: \[\frac{acy - 3}{ac + 2cx}\] as an improper fraction.

Key concepts

Improper Fractions

When tackling algebraic expressions that contain fractions, it's crucial to understand improper fractions. An improper fraction is one where the numerator (the top part of the fraction) is greater than or equal to the denominator (the bottom part of the fraction). This might look counterintuitive because we often associate fractions with values less than one, but improper fractions represent numbers equal to or greater than one.

For example, in the expression \(\frac{7}{4}\), 7 is larger than 4, making it an improper fraction. The exercise we're considering asks for the final answer to be left as an improper fraction, which suggests that although such fractions can usually be converted into mixed numbers (a combination of a whole number and a proper fraction), the solution here should remain in fraction form, where the numerator is larger than or equal to the denominator. This is often useful in algebraic work, as it keeps the expressions tidy and easier to manipulate.

Common Denominator

Finding a common denominator is a pivotal step in the process of simplifying complex fractions. A common denominator refers to a shared multiple of the denominators of two or more fractions. Having a common denominator allows us to combine the fractions, which is necessary to simplify them.

In the given exercise, the common denominator is identified as \(ac\) for the terms within the numerator. By rewriting both terms over this common denominator, we can seamlessly combine them into a single fraction. This step is akin to finding a level playing field where different fractions can interact – imagine trying to compare prices in different currencies without converting them to a single standard. The concept of a common denominator does precisely this by converting all terms into a single 'currency' that can be easily compared and consolidated.

Factoring Algebraic Expressions

Factoring plays a vital role in simplifying algebraic expressions, including complex fractions. To factor an expression means to break it down into its simplest building blocks—like dismantling a Lego construction into individual bricks. In algebra, this typically involves finding terms or numbers that are in common between elements of a term and expressing them as a product of their factors.

During the simplification process in our exercise, factoring is used to simplify the numerator of the complex fraction. This is achieved by identifying \(x\) as a common factor in the terms of the numerator and then separating it from the rest of the expression. This kind of algebraic factoring not only makes simplification possible but also clarifies what can be canceled out when the expression is further simplified—much like removing duplicate pieces in a puzzle that do not affect the final image.