Problem 72 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. $$\frac{5}{3 n}-\frac{7}{4 m}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{20m - 21n}{12nm}\).

Step 1: Find the LCM of the Denominators

Identify the least common multiple (LCM) of the denominators \(3n\) and \(4m\). The LCM of \(3n\) and \(4m\) is \(12nm\) because \(3\) and \(4\) are coprime and the common multiple of \(n\) and \(m\) is \(nm\).

Step 2: Adjust the Fractions to Have a Common Denominator

Change the fractions so they both have the common denominator \(12nm\). For \(\frac{5}{3n}\): Multiply the numerator and the denominator by \(4m\) to get \(\frac{5 \times 4m}{3n \times 4m} = \frac{20m}{12nm}\).For \(\frac{7}{4m}\): Multiply the numerator and the denominator by \(3n\) to get \(\frac{7 \times 3n}{4m \times 3n} = \frac{21n}{12nm}\).

Step 3: Subtract the Fractions

Subtract the two fractions that now have a common denominator:\[ \frac{20m}{12nm} - \frac{21n}{12nm} = \frac{20m - 21n}{12nm} \]

Step 4: Simplify the Result if Possible

Check if the numerator and denominator have common factors that can be simplified. In this case, the numerator \(20m - 21n\) and the denominator \(12nm\) do not have any common factors, so the expression is already in its simplest form.

Key Concepts

Least Common Multiple (LCM)

When working with rational expressions, especially in operations like addition or subtraction, finding the Least Common Multiple (LCM) is crucial. It helps us combine terms by creating a common denominator. For the denominators given here, namely \(3n\) and \(4m\), the LCM is determined by looking at each component:

• The numbers 3 and 4 are prime to each other, meaning their product, 12, is their LCM.
• Variables \(n\) and \(m\), both in the denominator, multiply together to give \(nm\).

Bringing these factors together, the LCM of \(3n\) and \(4m\) becomes \(12nm\). Thus, the LCM allows us to rewrite multiple terms over a unified base, simplifying the process of subtraction or addition of fractions.
Understanding this foundational step ensures that combining or comparing rational expressions is efficient and clear.

Common Denominator

A critical part of simplifying fractions involves transforming each expression to have a common denominator. This step ensures that subtraction or addition occurs smoothly.
For the fractions \(\frac{5}{3n}\) and \(\frac{7}{4m}\), changing each to a common denominator of \(12nm\) is necessary:

• Multiply both the numerator and denominator of \(\frac{5}{3n}\) by \(4m\) to get \(\frac{20m}{12nm}\).
• For \(\frac{7}{4m}\), you multiply by \(3n\), giving \(\frac{21n}{12nm}\).

Having a shared denominator of \(12nm\) allows the straightforward subtraction of the fractions. Achieving a common denominator standardizes the fractions, facilitating the upcoming step of operation.

Simplifying Fractions

After obtaining a common denominator, simplifying the result is the final step. The subtraction of our adjusted fractions \(\frac{20m}{12nm} - \frac{21n}{12nm}\) results in \(\frac{20m - 21n}{12nm}\).
Once the fraction is in this form, it's vital to check if it can be simplified further.
Simplification involves factoring both the numerator and the denominator. If they share common factors, they can be reduced. In this case, the numerator \(20m - 21n\) and the denominator \(12nm\) do not share any common factors. Thus, the expression remains as is. Simplifying fractions reduces them to their least complex form, ensuring clarity and efficiency in calculations.