Problem 49 - Algebra for College Students

Question

Convert each rational expression into an equivalent rational expression that has the indicated denominator. $$ \frac{1}{5}, \frac{?}{50} $$

Step-by-Step Solution

Answer

The equivalent rational expression is \( \frac{10}{50} \).

Step 1: Identify the given rational expression and the desired denominator

The given rational expression is \( \frac{1}{5} \) and the desired denominator is \( 50 \).

Step 2: Determine the factor needed to adjust the denominator

To convert \( \frac{1}{5} \) to an equivalent expression with a denominator of 50, we need to find the factor that relates 5 to 50. This can be done by dividing the desired denominator by the original denominator: \(\frac{50}{5} = 10\).

Step 3: Adjust the numerator and denominator of the original expression

To adjust the original fraction, multiply both the numerator and the denominator by the factor determined in Step 2. Thus, \( \frac{1}{5} \) becomes \( \frac{1 \times 10}{5 \times 10} \).

Step 4: Simplify the expression

\( \frac{1 \times 10}{5 \times 10} = \frac{10}{50} \). This is the expression with the desired denominator.

Key Concepts

Fraction Conversion

Understanding fraction conversion is important when working with rational expressions. Fraction conversion involves transforming a fraction into an equivalent fraction with a different denominator or numerator. This is particularly useful when you need to compare fractions, add or subtract them, or solve certain equations.
For example, let's convert the fraction \(\frac{1}{5}\) to one with a denominator of 50. The goal is to maintain the fraction's value while changing its format.
The key is to use multiplication or division to change the denominator. Find what number you need to multiply or divide the original denominator by to get the desired denominator. In our case, we multiply 5 by 10 to get 50. Therefore, we also need to multiply the numerator (1) by the same number (10).
This gives us \(\frac{1 \times 10}{5 \times 10} = \frac{10}{50}\). Both \(\frac{1}{5}\) and \(\frac{10}{50}\) are equivalent fractions.

Common Denominators

Common denominators are necessary when adding or subtracting fractions. They refer to the same bottom number (denominator) in different fractions.
To compare or operate on fractions easily, first find a common denominator.
For example, consider the fractions \(\frac{1}{5}\) and \(\frac{2}{7}\). Their denominators (5 and 7) are different. To add or subtract these fractions, we must find a common denominator.
The least common denominator (LCD) of 5 and 7 is 35.
To make both fractions have this denominator, convert each by multiplying the numerator and denominator by the same number: \(\frac{1}{5} = \frac{1 \times 7}{5 \times 7} = \frac{7}{35}\) and \(\frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35}\).
Now, we can add or subtract these fractions because they share a common denominator.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. A fraction is considered simplified if the numerator and denominator have no common factors other than 1.
For example, the fraction \(\frac{10}{50}\) can be simplified. Both 10 and 50 have a common factor of 10. \(\frac{10 \text{ divided by } 10}{50 \text{ divided by } 10} = \frac{1}{5}\)
This makes \(\frac{1}{5}\) the simplest form of \(\frac{10}{50}\).
Simplifying fractions helps in comparing them and makes calculations easier.
Here are the steps to simplify a fraction:

• Find the Greatest Common Divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

Practice this with different fractions to get comfortable with the process.