Problem 11 - Prealgebra

Question

Find the least common denominator (LCD) of each pair of fractions. $$\frac{10}{45} \circ \frac{2}{9}$$

Step-by-Step Solution

Answer

The least common denominator of the fractions is 45.

Step 1: Identify the Denominators

In the given fractions \( \frac{10}{45} \) and \( \frac{2}{9} \), the denominators are \( 45 \) and \( 9 \).

Step 2: Factorize Each Denominator

Factorize each denominator into its prime factors. \( 45 = 3^2 \times 5 \) and \( 9 = 3^2 \).

Step 3: Identify the Highest Power of Each Prime Factor

For each prime factor found in the denominators, determine the highest power. The prime factor \( 3 \) appears as \( 3^2 \) and the prime factor \( 5 \) appears as \( 5^1 \).

Step 4: Calculate the Least Common Denominator

The least common denominator is the product of the highest powers of all prime factors. Therefore, the LCD is \( 3^2 \times 5 = 45 \).

Key Concepts

Fractions

Working with fractions involves numbers expressed in the form of a ratio or a division of two numbers, where the numerator is divided by the denominator. In the example \( \frac{10}{45} \), \( 10 \) is the numerator, and \( 45 \) is the denominator. For \( \frac{2}{9} \), \( 2 \) is the numerator and \( 9 \) is the denominator. Fractions can represent parts of a whole, and understanding them is crucial when comparing or performing operations like addition, subtraction, multiplication, and division. When adding or subtracting fractions, a common denominator is required. This practice ensures that the fractions are on the same fraction line, making operations between different denominators feasible.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks - prime numbers. A prime number is any whole number greater than 1 that has no divisors other than 1 and itself. To find the prime factorization of a number:

• Start dividing the number by the smallest prime number, \( 2 \). If it isn't divisible by \( 2 \), try the next prime numbers, which are \( 3, 5, 7, \) and so on.
• Continue the division until the quotient is \( 1 \). At each step, record the prime numbers used for division.

For example, the prime factorization of \( 45 \) is \( 3^2 \times 5 \). Similarly, \( 9 \) can be factorized as \( 3^2 \). Prime factorization is crucial in finding the least common denominator, as evident in the exercise.

Denominators

Denominators are key when dealing with fractions as they define the number of equal parts that make up a whole. In fractions \( \frac{10}{45} \) and \( \frac{2}{9} \), \( 45 \) and \( 9 \) are denominators, respectively. Understanding and comparing denominators are essential for operations involving fractions, especially addition and subtraction. To perform such operations:

• Identify the denominators of each fraction involved.
• Find the least common denominator (LCD), which is the smallest number that both denominators can divide evenly into.

The LCD enables fractions to be expressed with the same denominator, simplifying addition or subtraction. In our example, we found the LCD to be \( 45 \) by using prime factorization to identify the highest powers of each factor involved. This allows for an easy adjustment of fractions, facilitating straightforward mathematical operations.