Problem 46 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{9}{r}-\frac{2}{3}$$

Step-by-Step Solution

Answer

\( \frac{27 - 2r}{3r} \)

Step 1: Find the Least Common Denominator (LCD)

The denominators of the fractions are \( r \) and \( 3 \). The least common denominator (LCD) for these fractions is the product of the two denominators since they are coprime. Thus, the LCD is \( 3r \).

Step 2: Adjust Each Fraction to the Common Denominator

To combine the fractions, rewrite each fraction with the LCD as the denominator. For \( \frac{9}{r} \), multiply both the numerator and the denominator by \( 3 \) to get \( \frac{27}{3r} \). For \( \frac{2}{3} \), multiply both the numerator and the denominator by \( r \) to get \( \frac{2r}{3r} \).

Step 3: Perform the Subtraction

Now that both fractions have the same denominator, subtract them: \( \frac{27}{3r} - \frac{2r}{3r} = \frac{27 - 2r}{3r} \).

Step 4: Simplify (if necessary)

Check if the fraction can be simplified further. In this case, \( \frac{27 - 2r}{3r} \) is already in its simplest form as there are no common factors in the numerator and the denominator.

Key Concepts

Least Common Denominator

When dealing with fractions with different denominators, solving for the least common denominator (LCD) is a critical step for addition or subtraction. The LCD is essentially the smallest number into which both of the denominators can divide evenly.

To find the LCD of two numbers, you might consider their factors:

• If the denominators share no common factors other than 1, they are referred to as coprime. In this situation, like the denominators \( r \) and \( 3 \) in the given problem, the LCD is simply the product of those coprime denominators.
• If the denominators do have common factors, the LCD will be the least common multiple (LCM) of the two denominators.

Recognizing and calculating LCD allows you to rewrite the fractions with a common denominator, which is the first step towards performing operations like addition and subtraction on rational expressions.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator are as small as possible.

• To simplify a fraction, you have to divide both the numerator and the denominator by their greatest common divisor (GCD).
• The goal is to have the fraction in such a way that no number except 1 can evenly divide both, making it easier to interpret and less cumbersome in calculations.
• Think of it like cutting a pizza into fewer slices, where each slice is bigger but still represents the same total quantity.

In our example, after performing the subtraction \( \frac{27}{3r} - \frac{2r}{3r} = \frac{27 - 2r}{3r} \), we checked if it could be simplified further. Since \( 27 - 2r \) and \( 3r \) share no common factors, the fraction is already in its simplest form.

Coprime Denominators

Two numbers are coprime if they have no common factors other than 1, meaning their greatest common divisor (GCD) is 1.

• In mathematics, having coprime denominators simplifies finding the least common denominator because their LCD is just the product of the denominators.
• This was the case in the example with \( r \) and \( 3 \); multiplying them gives the least common denominator \( 3r \).

Understanding coprime numbers is handy when simplifying complex rational expressions. It often requires less computation to arrive at the combined fractions, thereby making algebraic operations more straightforward and efficient.