Problem 7 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To build \(\frac{x}{x+2}\) so that it has a denominator of \(5(x+2),\) we multiply it by 1 in the form of _____.

Step-by-Step Solution

Answer

Multiply by \( \frac{5}{5} \).

Step 1: Identifying Desired Denominator

The original expression is \( \frac{x}{x+2} \). We need a new expression with a denominator of \( 5(x+2) \).

Step 2: Formulating Equivalent of One

To change the denominator from \( x+2 \) to \( 5(x+2) \), we note that we can multiply \( \frac{x}{x+2} \) by some fraction that equals 1. This fraction should be \( \frac{5}{5} \) because it equals 1 and introduces a 5 into the denominator.

Step 3: Multiplying Fraction by Equivalent of One

Multiply the original fraction by \( \frac{5}{5} \) to build the fraction so that the denominator matches the desired form: \( \frac{x}{x+2} \times \frac{5}{5} = \frac{5x}{5(x+2)} \).

Step 4: Verifying New Expression

Check that the new expression \( \frac{5x}{5(x+2)} \) has the desired denominator of \( 5(x+2) \), confirming that our equivalent of one \( \frac{5}{5} \) was correctly chosen.

Key Concepts

Equivalent Fractions

When we talk about equivalent fractions, we're referring to fractions that may look different but represent the same value.
In mathematical terms, if you multiply both the numerator (top part) and the denominator (bottom part) of a fraction by the same non-zero number, you create a fraction equivalent to the original one.
This principle is central to many algebra problems because it allows us to transform fractions in a way that is more useful for specific calculations or comparisons.

• For example, consider the fraction \( \frac{1}{2} \). If we multiply the numerator and denominator by 3, we get \( \frac{3}{6} \), which has the same value as \( \frac{1}{2} \) because 3 divided by 6 equals 0.5, just like 1 divided by 2.
• In algebraic fractions, we apply this concept to manipulate expressions while preserving their equivalence.

Denominator Manipulation

Denominator manipulation is a technique used to adjust the denominator of a fraction without changing its overall value.
This allows us to rewrite fractions so that they have a specific desired denominator, which can simplify comparison, addition, or subtraction of fractions.

• To change the denominator of \( \frac{x}{x+2} \) to \( 5(x+2) \), we multiply by a fraction that equals 1, such as \( \frac{5}{5} \).
• By multiplying \( \frac{x}{x+2} \) by \( \frac{5}{5} \), we introduce 5 into the denominator without changing the fraction’s value, producing \( \frac{5x}{5(x+2)} \).

This manipulation is handy when solving equations involving fractions or when regrouping terms to facilitate operations.

Multiplying Fractions

Multiplying fractions is a straightforward process that involves multiplying the numerators together and the denominators together.
This process keeps the fraction consistent and equivalent to the original fractions before multiplication.

• For example, when multiplying \( \frac{a}{b} \) by \( \frac{c}{d} \), the resulting fraction is \( \frac{ac}{bd} \).
• The key to maintaining the value is ensuring any number they are multiplied by is equivalent to 1, such as \( \frac{5}{5} \) in our scenario.

For our specific problem, multiplying \( \frac{x}{x+2} \) by \( \frac{5}{5} \) yields \( \frac{5x}{5(x+2)} \).
We have only multiplied by 1 in a different form to maintain the equivalence of the original expression.