Problem 62 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \frac{7}{30}+\frac{3}{18} $$

Step-by-Step Solution

Answer

The sum of the fractions is \(\frac{2}{5}\).

Step 1: Identify the Denominators

First, we need to identify the denominators of the fractions involved, which are 30 and 18.

Step 2: Find the Least Common Denominator (LCD)

To add the fractions, we need a common denominator. We find the Least Common Multiple (LCM) of 30 and 18. Factor 30 as \(30 = 2 \times 3 \times 5\) and 18 as \(18 = 2 \times 3^2\). The LCM is \(2 \times 3^2 \times 5 = 90\). Thus, the LCD is 90.

Step 3: Adjust the Fractions to Have the Same Denominator

Convert each fraction to an equivalent fraction with a denominator of 90. For \(\frac{7}{30}\), multiply both the numerator and the denominator by 3, giving \(\frac{21}{90}\). For \(\frac{3}{18}\), multiply both numerator and denominator by 5, resulting in \(\frac{15}{90}\).

Step 4: Add the Fractions

Now that both fractions have the same denominator, add them: \(\frac{21}{90} + \frac{15}{90} = \frac{36}{90}\).

Step 5: Simplify the Resulting Fraction

Simplify \(\frac{36}{90}\) by finding the greatest common divisor (GCD) of 36 and 90, which is 18. Divide both the numerator and the denominator by 18: \(\frac{36}{90} = \frac{2}{5}\).

Key Concepts

Least Common Denominator

When adding fractions, it's crucial to work with the same denominator. This makes the process of adding them straightforward. To find the least common denominator (LCD), determine the least common multiple (LCM) of the fractions' denominators. In the exercise, the denominators were 30 and 18.

Start by breaking down the denominators into their prime factors:

• 30 becomes \(2 \times 3 \times 5\)
• 18 becomes \(2 \times 3^2\)

The LCM is found by taking the highest power of each prime number found: \(2\), \(3^2\), and \(5\). Multiply these together to get 90.

Thus, the LCD is 90, allowing us to express both fractions with this common denominator, making adding them possible.

Simplifying Fractions

After finding the least common denominator and performing the addition, the next step is to simplify the resulting fraction. Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than one. This process makes it easier to understand and use the fraction in calculations.

For instance, after adding the fractions, you got \(\frac{36}{90}\). Both 36 and 90 have common factors, so we can simplify this fraction. To simplify it, divide both the numerator and the denominator by their greatest common divisor (GCD).

This results in the simplified form \(\frac{2}{5}\), a cleaner and more usable figure.

Greatest Common Divisor

A key part of fraction simplification is finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without a remainder. This is essential in reducing fractions to their simplest form.

To find the GCD, list out the factors of each number until you identify the largest common one. For 36 and 90, the divisors are:

• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

The largest common factor is 18.

Therefore, 18 is the GCD of 36 and 90. Using the GCD, you simplify \(\frac{36}{90}\) to \(\frac{2}{5}\) by dividing both the numerator and denominator by 18, achieving the simplest form possible.