Problem 59 - Algebra 1: Concepts and Skills

Question

Find the least common denominator of the pair of fractions. $$\frac{2}{3}, \frac{4}{7}$$

Step-by-Step Solution

Answer

The least common denominator of the fractions \( \frac{2}{3} \) and \( \frac{4}{7} \) is 21.

Step 1: Identify the Denominators

The denominators of the two fractions, \( \frac{2}{3} \) and \( \frac{4}{7} \), are 3 and 7, respectively.

Step 2: Find the Least Common Multiple (LCM)

Since 3 and 7 are both prime numbers and have no common factors other than 1, their least common multiple (LCM) is obtained by directly multiplying them. Therefore, LCM of 3 and 7 is \( 3 * 7 = 21 \).

Step 3: Confirm the Least Common Denominator

Our Least Common Denominator (LCD) is the LCM of the denominators which is 21. Therefore, the LCD of \( \frac{2}{3} \) and \( \frac{4}{7} \) is 21. This is the smallest positive integer that both 3 and 7 divides into exactly.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. A fraction consists of two main parts: the numerator and the denominator. The numerator is the top number and indicates how many parts are being considered, while the denominator, the bottom number, indicates the total number of equal parts the whole is divided into. For instance, in the fraction \( \frac{2}{3} \), 2 is the numerator and 3 is the denominator.
Fractions can be used to express numbers that are not whole, like one half or three quarters. Knowing how to work with fractions, such as adding or comparing them, often requires finding a common denominator, which brings us to the concept of least common denominators.

Finding the Least Common Multiple (LCM)

The Least Common Multiple, or LCM, is an essential concept when working with fractions, particularly when adding or comparing them. It's the smallest number that is a multiple of each of the denominators you're working with. To find the LCM of two or more numbers, you can use several methods:

• **Prime Factorization**: Break down each number into its prime factors and select the highest power of each factor present.
• **Listing Multiples**: List the multiples of each number until you find the smallest one that appears in all lists.
• **Direct Multiplication**: If the numbers are co-prime (like 3 and 7 in our example), their product is the LCM.

As in our exercise, where we had to find the LCM of 3 and 7, we see they are both prime numbers with no common factors other than 1. Hence, we directly multiplied them to get an LCM of 21.

Understanding Denominators in Fractions

The denominator is crucial in fractions because it determines the fraction's size and how it relates to other fractions. It's important to know that the larger the denominator, the smaller each piece is, because the whole is divided into more parts.
When working with fractions that have different denominators, as in our problem with \( \frac{2}{3} \) and \( \frac{4}{7} \), it's necessary to find a common denominator to easily perform arithmetic operations like addition or subtraction. The **Least Common Denominator (LCD)** is the smallest common divisor of the denominators of two or more fractions. By using the LCD, fractions are reassembled into a form that makes them comparable or combinable.
In our example, the LCM of the denominators 3 and 7 provided us with the least common denominator, which is 21. This allows the fractions to be rewritten as fractions with the same denominator and further manipulated as needed.