Problem 40 - Elementary Algebra

Question

For exercises \(35-86\), simplify. $$ \frac{6 w}{25}-\frac{3 w}{10} $$

Step-by-Step Solution

Answer

\frac{-3 w}{50}

Step 1: Identify the denominators

The given fractions are \(\frac{6 w}{25} - \frac{3 w}{10}\). Notice that the denominators are 25 and 10.

Step 2: Find the Least Common Denominator (LCD)

To combine the fractions, we need a common denominator. The least common multiple of 25 and 10 is 50. Therefore, the LCD is 50.

Step 3: Rewrite each fraction with the LCD

Rewrite each fraction with 50 as the denominator. This means transforming \(\frac{6 w}{25}\) and \(\frac{3 w}{10}\) to have the denominator 50.

Step 4: Convert the first fraction

To convert \(\frac{6 w}{25}\) to a fraction with 50 as the denominator, multiply both the numerator and the denominator by 2. This gives us: \(\frac{6 w \times 2}{25 \times 2} = \frac{12 w}{50}\).

Step 5: Convert the second fraction

To convert \(\frac{3 w}{10}\) to a fraction with 50 as the denominator, multiply both the numerator and the denominator by 5. This gives us: \(\frac{3 w \times 5}{10 \times 5} = \frac{15 w}{50}\).

Step 6: Combine the fractions

Now that both fractions have the same denominator, we can subtract them: \(\frac{12 w}{50} - \frac{15 w}{50} = \frac{12 w - 15 w}{50}\).

Step 7: Simplify the numerator

Subtract the numerators: \(\frac{12 w - 15 w}{50} = \frac{-3 w}{50}\).

Key Concepts

Least Common Denominator

When dealing with fractions, it is essential to have a common denominator to perform operations like subtraction. The Least Common Denominator (LCD) is the smallest number that both denominators can divide into evenly.
For example, when working with fractions \(\frac{6 w}{25}\) and \(\frac{3 w}{10}\), you first identify the denominators, which are 25 and 10.
Then, find the least common multiple (LCM) of these numbers. In this case, the LCM of 25 and 10 is 50, hence the LCD is 50.
The LCD lets you rewrite the fractions with a common base, making operations like addition and subtraction easier.

Fraction Subtraction

Subtracting fractions involves expressing both fractions with the same denominator before performing the subtraction.
Using the fractions \(\frac{6 w}{25} - \frac{3 w}{10}\), you convert them to fractions with the LCD of 50.

• Multiply both the numerator and the denominator of the first fraction by 2: \(\frac{6 w \times 2}{25 \times 2} = \frac{12 w}{50}\).
• For the second fraction, multiply both the numerator and the denominator by 5: \(\frac{3 w \times 5}{10 \times 5} = \frac{15 w}{50}\).

This gives you two fractions with the same denominator: \(\frac{12 w}{50}\) and \(\frac{15 w}{50}\). Now you can subtract the numerators directly: \(\frac{12 w - 15 w}{50} = \frac{-3 w}{50}\).

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form.
After converting fractions to have a common denominator and performing the subtraction, simplify the resultant numerator if possible.
In the example of \(\frac{12 w - 15 w}{50}\), perform the subtraction in the numerator:

• Simplify the expression by subtracting numerators: \(\frac{12 w - 15 w}{50} = \frac{-3 w}{50}\).
• No further simplification is needed since -3w and 50 have no common factors other than 1.

So the simplified expression remains \(\frac{-3 w}{50}\). Simplifying expressions often makes them easier to understand and work with.

Numerator and Denominator

Understanding the terms 'numerator' and 'denominator' is crucial for working with fractions.

• The numerator is the top number in a fraction. It represents the number of parts you have.
• The denominator is the bottom number. It represents the total number of equal parts the whole is divided into.

In \(\frac{6 w}{25}\), \(6w\) is the numerator, and 25 is the denominator. To perform operations on fractions, you often need to adjust these parts.
For instance, converting \(\frac{6 w}{25}\) to have 50 as the denominator involves multiplying both the numerator and the denominator by 2: \(\frac{6 w \times 2}{25 \times 2} = \frac{12 w}{50}\). Both parts of the fraction must be handled simultaneously to maintain the value of the fraction.
Similarly, for \(\frac{3 w}{10}\), you multiply the numerator and denominator by 5 to get \(\frac{15 w}{50}\).