Problem 60 - Algebra 1: Concepts and Skills

Question

Find the least common denominator of the pair of fractions. $$\frac{1}{16}, \frac{9}{20}$$

Step-by-Step Solution

Answer

The least common denominator of \( \frac{1}{16} \) and \( \frac{9}{20} \) is 80.

Step 1: Prime Factorizations of Denominators

Express both denominators as the product of their prime factors: The prime factorization of 16: 2 * 2 * 2 * 2 = \(2^4\)The prime factorization of 20: 2 * 2 * 5 = \(2^2\) * 5

Step 2: Determine the LCD

To find the LCD, take the greatest power of each prime number from the prime factorizations of 16 and 20. So, the most 2 appeared in the prime factorizations was 4 times and the most 5 appeared was once. Therefore, the LCD is \(2^4\) * 5 = 16 * 5 = 80.

Key Concepts

Prime Factorization

Prime factorization is a vital concept in mathematics, especially when dealing with least common denominators or simplifying fractions. It involves breaking down a number into its basic building blocks—prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For example, to find the prime factorization of 16, we divide it by 2, which is a prime number, and continue dividing the results by 2 until we can't divide anymore. This gives us:

• 16 ÷ 2 = 8
• 8 ÷ 2 = 4
• 4 ÷ 2 = 2
• 2 ÷ 2 = 1

So, the prime factorization of 16 is 2 × 2 × 2 × 2 or \(2^4\). Similarly, we find the prime factors of 20 as 2 × 2 × 5 or \(2^2 \times 5\). Recognizing the prime factors of a number is crucial for operations like finding the least common denominator, simplifying fractions, or solving equations.

Fractions

Fractions represent parts of a whole. A fraction consists of two integers: a numerator and a denominator. The numerator appears above the line and indicates how many parts we are considering, while the denominator is below the line and reveals into how many equal parts the whole is divided.
When working with fractions, such as \(\frac{1}{16}\) and \(\frac{9}{20}\), a common task involves finding a common denominator to make calculations easier. This allows the fractions to be compared or added because the denominators—the number of equal parts—are the same.
By finding a common denominator, we ensure both fractions have the same baseline, essentially operating on similar grounds to make various operations possible. The least common denominator is particularly useful because it is the smallest number that can be used in place of the original denominators without affecting the fractions' values.

Greatest Power of Prime Numbers

When finding the least common denominator, using the greatest power of each prime number from the factorization of each denominator is key. This ensures the resulting denominator is the smallest one that both fractions can hypothetically "fit" into.
For instance, in our case of \(\frac{1}{16}\) and \(\frac{9}{20}\), the relevant prime factors are 2 and 5. From the prime factorization:

• 16 (\(2^4\)) utilizes the prime 2 to the fourth power.
• 20 (\(2^2 \times 5\)) utilizes 2 to the second power and 5 to the first power.

The greatest power of 2 from these is \(2^4\), and the greatest power of 5 is \(5^1\). Hence, the least common denominator becomes \(2^4 \times 5 = 80\). This approach, focusing on the greatest power of each prime factor, ensures that the least common denominator will suffice for both fractions and is not unnecessarily large.