Problem 30 - Prealgebra

Question

Find the least common denominator (LCD) of each pair of fractions. $$\frac{1}{4}, \frac{7}{8}$$

Step-by-Step Solution

Answer

The least common denominator of \( \frac{1}{4} \) and \( \frac{7}{8} \) is 8.

Step 1: Identify the Denominators

The denominators of the given fractions are 4 and 8.

Step 2: Find the Least Common Multiple (LCM)

To find the least common denominator, we need to find the least common multiple of 4 and 8. Write down the multiples: - Multiples of 4: 4, 8, 12, 16, 20, ... - Multiples of 8: 8, 16, 24, 32, ... The smallest multiple common to both lists is 8.

Step 3: Confirm the LCD

Verify that 8 is the smallest number that is a multiple of both 4 and 8. Since the least common multiple of 4 and 8 is indeed 8, this means the least common denominator of the fractions is also 8.

Key Concepts

Understanding Fractions

When we talk about fractions, we're describing a way to express parts of a whole. A fraction consists of two parts:

• The numerator, which is the top number, represents how many parts we have.
• The denominator, which is the bottom number, shows the total number of equal parts the whole is divided into.

For example, in the fraction \(\frac{1}{4}\), the numerator is 1, and the denominator is 4. This means you have 1 part out of a total of 4 parts. Fractions are a fundamental concept because they allow us to express quantities that are not whole numbers.
Understanding how to manipulate and compare fractions is important, and finding a common denominator is often a crucial step in this process.

Finding the Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest number that two or more numbers divide into evenly. When dealing with fractions, we often need to find the LCM of the denominators, which helps us convert the fractions to have the same denominator. This is known as finding the Least Common Denominator (LCD).
To find the LCM, we list the multiples of each denominator and identify the smallest multiple they share. For example:

• Multiples of 4 are 4, 8, 12, 16, 20, and so on.
• Multiples of 8 are 8, 16, 24, 32, and so on.

The smallest common multiple between 4 and 8 is 8. Thus, 8 is the LCM of 4 and 8, making it the least common denominator for the fractions \(\frac{1}{4}\) and \(\frac{7}{8}\).
Finding the LCM enables us to work more easily with the fractions in calculations such as addition and subtraction.

Understanding Denominators

The denominator in a fraction indicates into how many equal parts the whole is divided. It tells us the total number of pieces that make up one whole unit. Observing the denominator is key when comparing fractions or performing calculations like addition and subtraction.
When working with fractions, different denominators can make adding or subtracting them directly challenging. For instance, if we have the fractions \(\frac{1}{4}\) and \(\frac{7}{8}\), their denominators are different. To solve calculations involving these fractions:

• Find a common denominator, like we did with the LCM, to ensure the fractions have the same bottom number.
• The common denominator is derived from the LCM of the original denominators.

Once all fractions share the same denominator, they are easier to manage and manipulate in equations. This process of finding a shared denominator allows us to perform arithmetic operations consistently across different fractions.