Problem 24 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{2 n+1}{9}-\frac{n+3}{12} $$

Step-by-Step Solution

Answer

\( \frac{5(n-1)}{36} \)

Step 1: Identify the common denominator

To subtract the fractions \( \frac{2n+1}{9} \) and \( \frac{n+3}{12} \), we need a common denominator. The denominators are 9 and 12. The least common multiple (LCM) of 9 and 12 is 36.

Step 2: Convert fractions to have the common denominator

Rewrite each fraction with the common denominator 36. For \( \frac{2n+1}{9} \), multiply both the numerator and the denominator by 4 to get \( \frac{4(2n+1)}{36} \). For \( \frac{n+3}{12} \), multiply both the numerator and the denominator by 3 to get \( \frac{3(n+3)}{36} \).

Step 3: Rewrite the expressions

Now write the modified fractions: \( \frac{8n+4}{36} - \frac{3n+9}{36} \).

Step 4: Subtract the numerators

Since the denominators are the same, you can subtract the numerators directly: \( (8n + 4) - (3n + 9) = 8n + 4 - 3n - 9 = 5n - 5 \). Write the result as a single fraction: \( \frac{5n - 5}{36} \).

Step 5: Simplify the fraction

Factor the numerator to see if further simplification is possible: \( 5n - 5 = 5(n - 1) \). The fraction becomes \( \frac{5(n-1)}{36} \). Since there are no common factors between the numerator and the denominator, this is the simplest form.

Key Concepts

Common Denominator

Understanding what a "common denominator" is, plays a crucial role when working with rational expressions such as fractions, especially when you want to add or subtract them. A common denominator is a shared multiple of the denominators of two or more fractions.
To subtract or add fractions effectively, their denominators must be the same. This makes it easy to compare them directly. When two fractions do not have the same denominator, you must find a common one to make them compatible for direct operations.
In the exercise provided, the fractions \( \frac{2n+1}{9} \) and \( \frac{n+3}{12} \) have different denominators (9 and 12). Using a common denominator, in this case, the Least Common Multiple of these numbers, allows the fractions to be rewritten for straightforward subtraction.

Least Common Multiple (LCM)

The least common multiple (LCM) is helpful when dealing with rational expressions because it is the smallest number that both denominators (or any given numbers) divide into evenly. Finding the LCM is essential because it can serve as the common denominator when adding or subtracting fractions.
In our exercise, the denominators given are 9 and 12. To find the LCM, begin by listing out a few multiples of each number:

• Multiples of 9: 9, 18, 27, 36, 45...
• Multiples of 12: 12, 24, 36, 48...

The LCM is the first multiple that appears on both lists, which is 36 in this instance.
Knowing the LCM simplifies rewriting each fraction with a common denominator, making subtraction or addition possible and much easier.

Simplifying Fractions

After performing operations like addition or subtraction on rational expressions, the resulting fraction should be simplified for clarity and precision. Simplifying fractions involves reducing them to their lowest terms. This means eliminating any common factors between the numerator and the denominator, except for 1.
In this exercise, once you subtract the numerators and combine the expressions into one fraction \( \frac{5n - 5}{36} \), you will look for common factors. The expression \( 5n - 5 \) can be factored out as \( 5(n - 1) \).
Since 36 and 5 have no common factors apart from 1, the fraction is already in its simplest form, \( \frac{5(n-1)}{36} \). Remember, a fraction is considered simplified when there are no more numbers, apart from 1, that can both divide the numerator and the denominator.