Problem 35 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. See Example 3 . $$\frac{11}{5 m}-\frac{5}{6 m}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{41}{30m} \).

Step 1: Find a Common Denominator

To perform the subtraction, both fractions must have the same denominator. Given two fractions \( \frac{11}{5m} \) and \( \frac{5}{6m} \), we need a common denominator. To find it, calculate the least common multiple of the denominators \( 5m \) and \( 6m \). This gives us a common denominator of \( 30m \).

Step 2: Rewrite Each Fraction

Rewrite each fraction with the common denominator of \( 30m \). To convert \( \frac{11}{5m} \) to a fraction with denominator \( 30m \), multiply both the numerator and denominator by 6, giving \( \frac{66}{30m} \). Similarly, convert \( \frac{5}{6m} \) by multiplying both the numerator and denominator by 5, resulting in \( \frac{25}{30m} \).

Step 3: Subtract the Fractions

With the common denominator, subtract the two fractions: \( \frac{66}{30m} - \frac{25}{30m} \). Subtract the numerators while keeping the denominator the same: \( \frac{66 - 25}{30m} = \frac{41}{30m} \).

Step 4: Simplify the Fraction

Check if the fraction \( \frac{41}{30m} \) can be simplified further. Since 41 is a prime number and doesn't divide evenly into 30, the fraction is already in its simplest form.

Key Concepts

Common Denominator

When working with algebraic fractions, finding a common denominator is crucial to perform operations like addition and subtraction. A common denominator is essentially a shared multiple of the denominators involved. This allows different fractions to be expressed in a form that makes it easier to combine them.
For example, suppose you have the fractions \( \frac{11}{5m} \) and \( \frac{5}{6m} \). To find the common denominator, you must find the least common multiple (LCM) of the two denominators \( 5m \) and \( 6m \).

• Start by considering the numeric parts: 5 and 6.
• The LCM of 5 and 6 is 30 since this is the smallest number divisible by both.
• Next, consider the variable "m", which is already a common factor. Thus, the LCM is \( 30m \).

With this common denominator, you can now rewrite each fraction so they share the same base, making subtraction (or addition) possible.

Subtracting Fractions

Once you have fractions with the same denominator, subtracting them is straightforward. This process involves dealing with only the numerators:

• Using our common denominator \( 30m \), rewrite the original fractions.
• \( \frac{11}{5m} \) becomes \( \frac{66}{30m} \) and \( \frac{5}{6m} \) becomes \( \frac{25}{30m} \).
• Subtract the numerators by finding the difference: \( 66 - 25 = 41 \).
• Keep the common denominator \( 30m \): \( \frac{41}{30m} \).

During subtraction, remember:

• Numerators are directly subtracted: be careful about signs when negatives are involved.
• The denominator remains constant and does not change.

This methodology simplifies working with fractions and is similar to handling simple numerical fractions, just with variables involved.

Fraction Simplification

After performing operations with fractions, always check if the resulting fraction can be simplified. Simplification reduces fractions to their simplest form, making them easier to interpret.
The fraction \( \frac{41}{30m} \) needs to be checked:[list]

• Examine the numerator and denominator for any common factors.
• 41 is a prime number, meaning its only divisors are 1 and itself.
• Since 41 doesn't divide 30 evenly, and they have no common factors, the fraction is already as simple as it gets.

Understanding simplification helps enhance clarity in mathematical solutions and ensures that results are presented in the most elegant and understandable form. Always verify if any part of a fraction can be reduced to improve mathematical communication.