Problem 30 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{5}{3 a}-\frac{3}{4 a}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{11}{12a}\).

Step 1: Identify common denominator

To subtract two fractions, they must have the same denominator. In this case, the denominators are \(3a\) and \(4a\). The common denominator is \(12a\).

Step 2: Rewrite the fractions

Rewrite the problem so both fractions have the common denominator. The first fraction, when rewritten with the new denominator will be \(\frac{20}{12a}\) and the second fraction will be \(\frac{9}{12a}\). The problem can then be rewritten as \(\frac{20}{12a} - \frac{9}{12a}\).

Step 3: Subtract the fractions

Since now we have the same denominator, we can subtract the numerators directly: \(\frac{20-9}{12a}\) which yields \(\frac{11}{12a}\). Now we have a simplified expression.

Key Concepts

Common Denominator

When dealing with algebraic fractions, or any fractions for that matter, one of the key steps in operations such as addition or subtraction is to ensure that both fractions have the same denominator. This 'denominator' is the number at the bottom of the fraction. Suppose we have two fractions with differing denominators; we aim to make them the same to combine the fractions effectively.

To find a 'common denominator', you typically identify the least common multiple (LCM) of the existing denominators. For example, if we have fractions with denominators of \(3a\) and \(4a\), the LCM is the smallest number that both can divide into without leaving a remainder.

• The LCM of \(3a\) and \(4a\) is \(12a\), as 12 is the smallest number both 3 and 4 can divide into.
• It's crucial to remember that we multiply both the numerator and denominator of each fraction by whatever it takes to get this new denominator.

This step sets the stage for subtraction by making the denominators compatible. It allows for a straightforward subtraction of the numerators in the next steps.

Subtracting Fractions

After finding a common denominator and rewriting the fractions accordingly, subtracting them becomes quite simple. Once both fractions have the same denominator, the process involves subtracting just the numerators while keeping the denominator constant.

Consider the rewritten fractions: \(\frac{20}{12a}\) and \(\frac{9}{12a}\). With the common denominator of \(12a\), the subtraction is straightforward.

• Subtract the second numerator from the first, meaning we calculate: \(20 - 9 = 11\).
• The new fraction is formed by placing this result over the common denominator: \(\frac{11}{12a}\).

The power of finding a common denominator earlier is evident here, as it shifts all the complexity to that one step, allowing for smooth sailing when subtracting the numerators.

Simplifying Fractions

Simplification is the final part of handling algebraic fractions. A fraction is simplified when the numerator and the denominator have no common factors other than one. This means breaking it down to its most reduced form.

In our case, after subtracting the fractions, we arrived at \(\frac{11}{12a}\), which thankfully, is already in its simplest form since:

• 11 and 12 have no common factors (11 is a prime number), and 12 does not share any factors with it.
• The term \(a\) in the denominator doesn’t affect this, as it doesn’t simplify with the numerator.

Always check for simplification in both numeric and algebraic terms. The goal is to ensure that the fraction is as simple as possible, making it easier to work with in subsequent mathematical operations.