Problem 43 - Algebra 2

Question

Simplify each expression. $$ \frac{11}{9}-\frac{7}{2 w}-\frac{6}{5 w} $$

Step-by-Step Solution

Answer

\(\frac{110w - 423}{90w}\) is the simplified expression.

Step 1: Find a common denominator

The fractions in the expression have denominators 9, 2w, and 5w. To simplify, we need a common denominator. The least common multiple of these denominators is 90w.

Step 2: Convert each fraction to have the common denominator

Convert each fraction to have the denominator 90w. For \(\frac{11}{9}\), multiply the numerator and the denominator by 10w to get \(\frac{110w}{90w}\). For \(\frac{7}{2w}\), multiply the numerator and the denominator by 45 to get \(\frac{315}{90w}\). For \(\frac{6}{5w}\), multiply the numerator and the denominator by 18 to get \(\frac{108}{90w}\).

Step 3: Subtract the fractions

Now that all the fractions have the same denominator, subtract them: \[\frac{110w}{90w} - \frac{315}{90w} - \frac{108}{90w} = \frac{110w - 315 - 108}{90w}\.\]

Step 4: Simplify the numerator

Combine the terms in the numerator: \(110w - 315 - 108 = 110w - 423\.\)

Step 5: Write the final simplified expression

The expression is \(\frac{110w - 423}{90w}\.\) This fraction cannot be simplified further since there are no common factors in the numerator and the denominator.

Key Concepts

Common Denominator

When working with fractions, especially when adding or subtracting them, it’s crucial to have a common denominator. A common denominator is a shared multiple of the denominators of the fractions involved. Imagine trying to subtract apples from oranges – it wouldn’t make sense without converting them into a common unit.

• A common denominator allows you to easily compare or operate on fractions.
• Think of it as creating a common language for different terms.

To find a common denominator for a set of fractions, we often search for the least common multiple (LCM) of the denominators. This approach ensures that the resulting calculations are less complex and easier to handle. By achieving a common denominator, you set the stage for seamless arithmetic operations between various rational expressions.

Least Common Multiple

The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of rational expressions, it's especially useful because it tells you the smallest value you can use as a common denominator for several fractions.
Example:

• In our problem, the denominators are 9, 2w, and 5w.
• We list the multiples of each denominator:
  9: 9, 18, 27, 36, 45...
  2w: 2w, 4w, 6w, 8w...
  5w: 5w, 10w, 15w...
• The smallest number that appears in all lists is 90w, making it the least common multiple.

The LCM helps simplify problems by providing a uniform base for dealing with multiple fractions efficiently. With 90w as our common denominator, we can rewrite each fraction with ease.

Numerator and Denominator Conversion

To perform operations like addition or subtraction on multiple fractions, we need to convert them to have a common denominator. This involves changing each fraction while keeping its overall value the same. We do this by multiplying the numerator and denominator by the same number, a process which is legal because it’s like multiplying by 1.

• Start with each fraction and determine what factor is needed to reach the common denominator.
• For \(rac{11}{9}\), multiply by \(10w\) to get \(rac{110w}{90w}\).
• For \(rac{7}{2w}\), multiply by \(45\) to get \(rac{315}{90w}\).
• Lastly, for \(rac{6}{5w}\), multiply by \(18\) to get \(rac{108}{90w}\).

Once converted, these fractions have the same denominator, and subtraction becomes straightforward. This conversion is key to ensuring each fraction can be easily combined in subsequent steps.

Simplification

After converting to a common denominator, you can perform the arithmetic necessary to simplify the expression. Simplification involves reducing fractions to their simplest forms. For our exercise, once each fraction shared a common denominator:

• Subtract the numerators: \(110w - 315 - 108\).
• The resulting numerator is \(110w - 423\).
• Place this over the common denominator \(90w\).

The final fraction, \(rac{110w - 423}{90w}\), is checked to see if further simplification is possible. Simplifying means dividing both the numerator and the denominator by any common factors.
In this case, there are no common factors, so our result represents the simplest form of the expression.