Problem 27 - Elementary and Intermediate Algebra

Question

Subtract and simplify the result, if possible. \(\frac{m-1}{6 m^{2}}-\frac{5}{6 m^{2}}\)

Step-by-Step Solution

Answer

The simplified result is \(\frac{m-6}{6m^2}\).

Step 1: Identify the Problem

The exercise asks us to subtract two fractions with the same denominator: \(\frac{m-1}{6m^2}\) and \(\frac{5}{6m^2}\). The common denominator makes the subtraction straightforward.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators directly: \((m-1) - 5 = m - 1 - 5 = m - 6\).

Step 3: Write the Resulting Fraction

Combine the result of the subtraction into one fraction: \(\frac{m - 6}{6m^2}\).

Step 4: Simplify the Fraction

Check if the fraction \(\frac{m - 6}{6m^2}\) can be simplified further. In this case, there are no common factors between the numerator and the denominator other than 1, so the fraction is already in its simplest form.

Key Concepts

Fraction Subtraction

Subtracting fractions may seem daunting, but it becomes simpler with a consistent strategy, especially when dealing with algebraic expressions. When the denominators of the fractions are identical, like in the given problem \(\frac{m-1}{6m^2}\) and \(\frac{5}{6m^2}\), you can subtract the numerators directly.

Here's how it works:

• Put the fractions over the same denominator.
• Subtract the second numerator from the first.

In this exercise, you subtract \(5\) from \(m-1\). This operation simplifies the equation to \(m-6\).

The hidden key is to make sure you perform the operations on the numerators accurately, while the common denominator remains unchanged.

Common Denominators

Understanding common denominators is crucial in fraction arithmetic. The denominator tells us "how many parts make up a whole" and must be the same when adding or subtracting fractions.

In our scenario, both fractions share the denominator \(6m^2\). This simplifies the subtraction as it allows us to focus solely on the numerators.

• Think of common denominators like the same-sized pieces of a pie.
• Maintains consistency, reducing the chance of errors.

Finding a common denominator isn't needed here because it's already provided, but in general problems, you'd find the least common multiple (LCM) of the denominators involved.

This understanding forms the foundation for smoothly subtracting or adding fractions in any algebraic context.

Simplifying Fractions

Simplifying a fraction means breaking it down to its simplest form by removing any common factors between the numerator and the denominator. It's a crucial last step that often reveals a simpler representation of the fraction.

In \(\frac{m-6}{6m^2}\), our goal was to check for any common factors.

• Search for common factors in both terms, starting from the smallest prime numbers.
• If no common factors other than 1 exist, the fraction is in its simplest form.

Here, since \(m-6\) and \(6m^2\) share no common factors, the given fraction is already as simplified as it can be.

This step is often skipped by beginners, but mastering it ensures a cleaner, more accurate representation of your results.