Problem 236 - Intermediate Algebra

Question

How do you find the LCD of 2 fractions?

Step-by-Step Solution

Answer

The LCD of \(\frac{3}{4}\) and \(\frac{5}{6}\) is 12.

Step 1: Identify the Denominators

Find the denominators of the two fractions. For example, in the fractions \(\frac{3}{4}\) and \(\frac{5}{6}\), the denominators are 4 and 6.

Step 2: Find the Prime Factors

Decompose each denominator into its prime factors. For 4: \(4 = 2 \times 2\). For 6: \(6 = 2 \times 3\).

Step 3: Identify Distinct Prime Factors

List all distinct prime factors from the decompositions. Here, the distinct prime factors are 2 and 3.

Step 4: Determine the Highest Powers

Find the highest power of each distinct prime factor. The highest power of 2 is \(2^2\) (from 4), and the highest power of 3 is \(3^1\) (from 6).

Step 5: Multiply the Highest Powers

Multiply the highest powers of each prime factor to find the LCD. \(2^2 \times 3 = 4 \times 3 = 12\). The LCD is 12.

Key Concepts

Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks, which are prime numbers. These are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the prime factors of 6 are 2 and 3, since 6 can be written as the product of these primes: 6 = 2 × 3. To find the prime factors of a number, you repeatedly divide the number by the smallest prime number until you are left with 1.

• Start with the smallest prime (2).
• If 2 divides the number, write down 2 and divide the number by 2.
• Continue with the next prime (3) and repeat.

Using prime factorization helps simplify other mathematical operations, such as finding the Least Common Denominator (LCD) of fractions.

Fractions

Fractions represent a part of a whole and consist of a numerator and a denominator. The numerator, the top number, indicates how many parts we have, while the denominator, the bottom number, shows the total number of equal parts the whole is divided into.

• For example, in the fraction \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator.
• Fractions can be compared, added, subtracted, multiplied, or divided.

To perform operations with different fractions, having a common denominator is essential, especially for addition and subtraction. That's where the Least Common Denominator (LCD) becomes useful.

Denominators

The denominator is the number below the fraction line that indicates the total number of parts the whole is divided into. When working with multiple fractions, finding a common denominator is crucial for various operations.
To find the Least Common Denominator (LCD), you follow these steps:

• Identify the denominators of the fractions.
• Use prime factorization to decompose each denominator into its prime factors.
• List all distinct prime factors.
• Determine the highest power of each distinct prime factor.
• Multiply these highest powers together to get the LCD.

For instance, for the fractions \(\frac{3}{4}\) and \(\frac{5}{6}\), the denominators are 4 and 6. Using the prime factorization and following the steps above: 4 = 2 × 2 and 6 = 2 × 3. The distinct prime factors are 2 and 3. Highest powers are 2² and 3¹. Multiplying these gives us the LCD = 4 × 3 = 12.