Problem 30 - Algebra and Trigonometry

Question

Perform the indicated operations. \(\frac{1}{4}+\frac{1}{5}\)

Step-by-Step Solution

Answer

\(\frac{1}{4} + \frac{1}{5} = \frac{9}{20}\) in simplest form.

Step 1: Identify the Least Common Denominator (LCD)

Before performing the addition, we need both fractions to have the same denominator. The denominators in the fractions \(\frac{1}{4}\) and \(\frac{1}{5}\) are 4 and 5, respectively. The Least Common Denominator (LCD) of 4 and 5 is 20.

Step 2: Convert Fractions to Equivalent Fractions

We convert each fraction to have the LCD as the denominator. - For \(\frac{1}{4}\), multiply the numerator and denominator by 5: \(\frac{1 \times 5}{4 \times 5} = \frac{5}{20}\).- For \(\frac{1}{5}\), multiply the numerator and denominator by 4: \(\frac{1 \times 4}{5 \times 4} = \frac{4}{20}\).

Step 3: Add the Fractions

Now that we have a common denominator, we can add the fractions: \(\frac{5}{20} + \frac{4}{20} = \frac{5 + 4}{20} = \frac{9}{20}\).

Step 4: Simplify the Result

Check if the resulting fraction \(\frac{9}{20}\) can be simplified. The greatest common divisor (GCD) of 9 and 20 is 1, so the fraction is already in its simplest form.

Key Concepts

Least Common Denominator

When you're adding fractions like \( \frac{1}{4} \) and \( \frac{1}{5} \), the first step is to find a common denominator, known as the Least Common Denominator (LCD). This ensures that you can accurately add the fractions. The LCD is the smallest number that both denominators (4 and 5, in this case) can divide into evenly.
To find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple they both share.

For denominators 4 and 5, their least common multiple is 20. Therefore, 20 is the LCD. This shared denominator will allow you to easily combine fractions by ensuring they have the same "size" unit fractionally.

Equivalent Fractions

Once you've found the LCD, the next step is to change each fraction into an equivalent fraction with this common denominator. Equivalent fractions are different fractions that represent the same portion of a whole. To convert \( \frac{1}{4} \) and \( \frac{1}{5} \) into equivalent fractions with a denominator of 20, follow these steps:
- For \( \frac{1}{4} \), multiply both the numerator and denominator by 5 (since \( 4 \times 5 = 20 \)). This gives you: \[ \frac{1 \times 5}{4 \times 5} = \frac{5}{20} \] - For \( \frac{1}{5} \), multiply both the numerator and denominator by 4 (since \( 5 \times 4 = 20 \)). This results in: \[ \frac{1 \times 4}{5 \times 4} = \frac{4}{20} \]These equivalent fractions are crucial because they allow you to add fractions directly by just adding their numerators.

Simplifying Fractions

After adding the fractions, it's important to see if the sum can be simplified to its simplest form. Simplifying fractions means reducing them so that the numerator and denominator have no common factors other than 1. In this case, after obtaining the sum \( \frac{9}{20} \), you should check for any common divisors.
To simplify:

• Determine the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

For \( \frac{9}{20} \), the GCD of 9 and 20 is 1. Since 1 is the only divisor, the fraction is already in its simplest form. This means \( \frac{9}{20} \) can't be reduced further. Simplifying fractions ensures clarity and keeps fractions as manageable as possible.