Problem 9 - Prealgebra

Question

Find the least common denominator (LCD) of each pair of fractions. $$\frac{3}{5}, \frac{5}{8}$$

Step-by-Step Solution

Answer

The least common denominator of \( \frac{3}{5} \) and \( \frac{5}{8} \) is 40.

Step 1: Identify the Denominators

The denominators are the numbers at the bottom of the fractions. For the fractions \( \frac{3}{5} \) and \( \frac{5}{8} \), the denominators are 5 and 8.

Step 2: Determine the Least Common Denominator

To find the least common denominator (LCD), we need to find the smallest number that both 5 and 8 can divide into evenly. This involves finding the least common multiple (LCM) of the denominators.

Step 3: List Multiples of Each Denominator

Write out some multiples for each denominator: - Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ... - Multiples of 8: 8, 16, 24, 32, 40, ...

Step 4: Identify the Smallest Common Multiple

From the lists of multiples, identify the smallest number that appears in both lists. The number 40 is the first common multiple in both lists, making it the least common multiple.

Step 5: State the Least Common Denominator

The least common denominator (LCD) of \( \frac{3}{5} \) and \( \frac{5}{8} \) is 40.

Key Concepts

Fractions

Fractions are an essential part of mathematics. They represent parts of a whole. A fraction consists of two numbers, one on top of the other, separated by a line.

The top number is called the numerator and represents how many parts we have. The bottom number is called the denominator and shows into how many equal parts the whole is divided.

For example, in the fraction \(\frac{3}{5}\), the numerator is 3 and the denominator is 5, indicating that the whole is divided into 5 parts and 3 parts are being considered.

Least Common Multiple

The least common multiple (LCM) is an important concept when dealing with fractions. It helps in finding the least common denominator (LCD) when adding or subtracting fractions with different denominators.

The LCM of two numbers is the smallest number that both numbers can divide evenly. For example, when determining the LCM of 5 and 8, we need to find a number that both 5 and 8 can divide without leaving a remainder.

Listing the multiples of each number can help to visually identify the LCM.

• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...
• Multiples of 8: 8, 16, 24, 32, 40, ...

The smallest number that appears in both lists, in this case, is 40. This makes 40 the least common multiple of 5 and 8.

Mathematical Reasoning

Mathematical reasoning involves thinking logically to find solutions to problems. It is especially useful in situations like finding the least common denominator (LCD) because it requires understanding relationships between numbers.

In solving fractional problems, you begin by identifying key elements such as the denominators. Using reasoning, you convert fractions to have the same denominator by using the least common multiple.
This process involves several steps:

• Identifying denominators
• Finding the LCM
• Using the LCM to adjust the fractions so they have a common denominator

By applying these logical steps, you arrive at a solution efficiently, showcasing the power of mathematical reasoning in solving fractional problems.