Problem 83 - Elementary Algebra

Question

For exercises \(35-86\), simplify. $$ \frac{1}{6 a}+\frac{2}{3 a}-\frac{3}{4 a} $$

Step-by-Step Solution

Answer

\[ \frac{1}{12a} \]

Step 1 - Identify the common denominator

First, identify the common denominator for the fractions. The denominators are 6a, 3a, and 4a. The least common multiple (LCM) of 6, 3, and 4 is 12. Therefore, the common denominator is 12a.

Step 2 - Rewrite each fraction with the common denominator

Rewrite each fraction so that they all have 12a as the denominator: \[\frac{1 \times 2}{6a \times 2} + \frac{2 \times 4}{3a \times 4} - \frac{3 \times 3}{4a \times 3}\] This simplifies to: \[ \frac{2}{12a} + \frac{8}{12a} - \frac{9}{12a} \]

Step 3 - Combine the fractions

With a common denominator, the fractions can be combined into a single fraction: \[ \frac{2 + 8 - 9}{12a} \] Simplify the numerator: \[ \frac{1}{12a} \]

Step 4 - Simplify the overall expression

Since the numerator and denominator cannot be simplified any further, the final simplified expression is \[ \frac{1}{12a} \]

Key Concepts

common denominator

Understanding the common denominator is crucial when working with fractions. The common denominator is a shared multiple of the denominators in different fractions. When you want to add or subtract fractions, they need to have the same denominator. This allows you to combine the fractions easily. For example, if you're given the fractions \(\frac{1}{6a}\), \(\frac{2}{3a}\), and \(\frac{3}{4a}\), the denominators are 6a, 3a, and 4a. To find the common denominator, you must determine the smallest number that all these denominators can go into evenly. This leads us to the concept of the least common multiple (LCM).

least common multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. Finding the LCM is a key step in simplifying algebraic fractions with different denominators. For instance, to find the LCM of 6, 3, and 4, you list the multiples of each number until you find the smallest common multiple.

• Multiples of 6: 6, 12, 18, 24,...
• Multiples of 3: 3, 6, 9, 12, 15,...
• Multiples of 4: 4, 8, 12, 16,...

As you can see, 12 is the smallest multiple that all the numbers share. Thus, the LCM of 6, 3, and 4 is 12. Once you find the LCM, you multiply each fraction by the necessary factors to create fractions with the same denominator. In this case, the common denominator for \(\frac{1}{6a}\), \(\frac{2}{3a}\), and \(\frac{3}{4a}\) becomes 12a.

fraction addition

When adding fractions, it is essential they have the same denominator. If they don’t, you must convert them so that they do. Here’s how you do it with our example:

• Rewrite \(\frac{1}{6a}\) as \(\frac{1 \times 2}{6a \times 2} = \frac{2}{12a}\)
• Rewrite \(\frac{2}{3a}\) as \(\frac{2 \times 4}{3a \times 4} = \frac{8}{12a}\)
• Rewrite \(\frac{3}{4a}\) as \(\frac{3 \times 3}{4a \times 3} = \frac{9}{12a}\)

Now that the fractions have the same denominator, you can add or subtract the numerators directly: \(\frac{2}{12a} + \frac{8}{12a} - \frac{9}{12a} = \frac{2 + 8 - 9}{12a} = \frac{1}{12a}\). It’s important to perform these operations carefully to avoid errors.

algebraic simplification

Algebraic simplification involves reducing expressions to their simplest form. This process makes it easier to understand and work with an equation. Once you have combined the algebraic fractions and performed the necessary additions or subtractions, you might get a result that can be further simplified.
If you have a fraction like \(\frac{1}{12a}\), you check if there are any common factors between the numerator and the denominator. In our example:

• The numerator is 1.
• The denominator is 12a.

Since 1 does not share any common factors with 12a, the expression is already in its simplest form. Thus, \(\frac{1}{12a}\) remains as is. Simplifying algebraic fractions improves your calculations and clarity in algebraic processes.