Problem 41 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Simplify the complex fraction. \(\frac{\frac{1}{2 x-5}-\frac{7}{8 x-20}}{\frac{x}{2 x-5}}\)

Step-by-Step Solution

Answer

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

Step 1: Find a common denominator

In order to subtract fractions, they must have the same denominator, a common denominator. The denominators are \(2x - 5\) and \(8x - 20\). It is easy to see that \(8x - 20\) is just \(4(2x - 5)\). So, we can rewrite the complex fraction as \(\frac{\frac{1}{2x - 5}-\frac{7}{4(2x - 5)}}{\frac{x}{2x - 5}}\)

Step 2: Subtract fractions in the numerator

We can now subtract the fractions in the numerator since they now have a common denominator: \(2x - 5\). This gives us: \(\frac{\frac{1 - 4}{2x - 5}}{\frac{x}{2x - 5}} = \frac{\frac{-3}{2x - 5}}{\frac{x}{2x - 5}}\)

Step 3: Simplify the complex fraction

The complex fraction simplification is done by multiplying the numerator by the reciprocal of the denominator, (since dividing by a fraction is equivalent to multiplying by its reciprocal). Hence, our fraction becomes \( \frac{-3}{2x - 5} \div \frac{x}{2x - 5} \). We multiply by the reciprocal of \( \frac{x}{2x - 5} \) which is \( \frac{2x - 5}{x} \), giving us \( \frac{-3}{2x - 5} \times \frac{2x - 5}{x} \). The \(2x - 5\) terms cancel out, leaving \( \frac{-3}{x} \).

Key Concepts

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When you are working with complex fractions, or any fractions that need to be added or subtracted, having a common denominator is essential. It's like giving all the fractions a uniform base so that they can be easily compared or combined.

Finding a common denominator involves looking at the denominators you have and determining the smallest number that each can divide into without a remainder. For example, if your denominators are 5 and 10, the common denominator is 10, because 5 goes into 10 exactly twice. However, the denominators in algebraic expressions, like the ones in our exercise \(2x - 5\) and \(8x - 20\), can also share a common factor, leading us to identify that \(8x - 20\) can be simplified to \(4(2x - 5)\), making \(2x - 5\) the common denominator.

Subtract Fractions

To subtract fractions, we must first ensure that they have a common denominator, as emphasized in an earlier section. Once we have established this, the numerators of the fractions can be directly subtracted from one another. This is like saying if you have a quarter of a pizza and you give away an eighth of a pizza, how much is left if both pieces come from pizzas of the same size?

In our exercise, we had two fractions \(\frac{1}{2x - 5} - \frac{7}{8x - 20}\) which, after finding a common denominator, allowed us to subtract the numerators to simplify the expression. It’s important to handle the subtraction carefully to avoid mistakes. This step simplifies the complex fraction significantly, reducing it to a single fraction in the numerator.

Reciprocal of a Fraction

The reciprocal of a fraction is simply a flipped version of the original fraction. Often referred to as the 'multiplicative inverse,' this means turning the numerator into the denominator and the denominator into the numerator. The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\), given that neither 'a' nor 'b' are zero, because division by zero is undefined.

In the context of our example, we used the reciprocal when simplifying the complex fraction by multiplying the numerator \(\frac{-3}{2x - 5}\) by the reciprocal of the denominator \(\frac{x}{2x - 5}\), which is \(\frac{2x - 5}{x}\). Multiplying by the reciprocal is essentially dividing by the original fraction, allowing the complex fraction simplification to be executed smoothly.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. These expressions represent quantities without a fixed value, often used to describe general rules or relationships.

In our problem, the algebraic expressions are represented by the denominators in the complex fraction, \(2x - 5\) and \(8x - 20\), and the variables within them complicate the arithmetic. Yet, understanding how to manipulate these expressions is key to simplifying the complex fraction. It's like having a recipe where the exact amount of an ingredient may change each time you bake, but the relationship between the ingredients remains the same.