Problem 12 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{y}{4}+\frac{3 y}{5} $$

Step-by-Step Solution

Answer

\( \frac{y}{4} + \frac{3y}{5} = \frac{17y}{20} \).

Step 1: Find the Least Common Denominator (LCD)

Identify the denominators of the given fractions: 4 and 5. The least common denominator of 4 and 5 is 20. This is because 20 is the smallest number that both 4 and 5 divide into evenly.

Step 2: Convert Fractions to Equivalent Fractions with LCD

Convert each fraction to have the same denominator. To change \( \frac{y}{4} \) to a denominator of 20, multiply both the numerator and denominator by 5: \( \frac{y \times 5}{4 \times 5} = \frac{5y}{20} \). Similarly, change \( \frac{3y}{5} \) to a denominator of 20 by multiplying both the numerator and denominator by 4: \( \frac{3y \times 4}{5 \times 4} = \frac{12y}{20} \).

Step 3: Add the Fractions

Now that the fractions have the same denominator, add them by combining their numerators: \( \frac{5y}{20} + \frac{12y}{20} = \frac{5y + 12y}{20} = \frac{17y}{20} \).

Step 4: Simplify Further if Needed

Check if the fraction \( \frac{17y}{20} \) can be simplified. Since 17 and 20 have no common factors other than 1, \( \frac{17y}{20} \) is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator

When we add fractions, it's important that they have the same denominator. The least common denominator (LCD) is the smallest number that both denominators can divide into without leaving a remainder. For example, if we have two fractions like \(\frac{y}{4}\) and \(\frac{3y}{5}\), their denominators are 4 and 5. The LCD of 4 and 5 is 20 because it is the smallest number that both 4 and 5 divide into evenly.
To find the LCD:

• List the multiples of each denominator.
• 4: 4, 8, 12, 16, 20, 24...
• 5: 5, 10, 15, 20, 25...

The first shared multiple is 20. Once the LCD is identified, you can proceed to convert each fraction so that they have this common denominator.

Simplifying Fractions

Simplification of fractions makes them easier to work with and comprehend. A fraction is simplified when the numerator and the denominator have no common divisors other than 1. Let's consider the result of our earlier operation, \(\frac{17y}{20}\). Here, both 17 and 20 have no common factors other than 1, meaning the fraction is already simplified.
To simplify a fraction:

• Check the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.

If our final answer is not in its simplest form, following the steps above will efficiently reduce the fraction without affecting its value.

Creating Equivalent Fractions

Equivalent fractions are different fractions that have the same value. For example, \(\frac{y}{4}\) can be converted to an equivalent fraction with a denominator of 20 by multiplying both its numerator and denominator by the same number. Here, we multiply by 5, resulting in \(\frac{5y}{20}\).
This process involves:

• Multiplying the numerator and the denominator by the same non-zero number. This keeps the overall value of the fraction unchanged.
• Ensuring you apply this to both fractions you are working with to maintain consistency in the operation.

Creating equivalent fractions is essential in operations involving fractions, as it aligns the denominators, allowing you to perform operations like addition and subtraction effectively.