Problem 32 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers or monomials. $$\frac{4}{5}, \frac{1}{2}$$

Step-by-Step Solution

Answer

The LCM of \(\frac{4}{5}\) and \(\frac{1}{2}\) is 4.

Step 1: Convert Fractions to Like Denominators

To work with fractions efficiently, find a common denominator. The denominators for \(\frac{4}{5}\) and \(\frac{1}{2}\) are 5 and 2, so the least common denominator (LCD) is 10. Convert \(\frac{4}{5}\) to \(\frac{8}{10}\) and \(\frac{1}{2}\) to \(\frac{5}{10}\).

Step 2: Determine LCM of Numerators

Now that the denominators are the same, focus on the numerators 8 and 5. Find their LCM. Since 8 and 5 have no common factors other than 1, their LCM is simply the product of the two numbers: \(8 \times 5 = 40\).

Step 3: Formulate the LCM of the Fractions

Put the LCM of the numerators over the common denominator to formulate the LCM of fractions: \[\frac{40}{10} = 4\].

Step 4: Simplify if Necessary

Finally, ensure the fraction is in its simplest form, if applicable. Since \(4\) is already an integer, it cannot be simplified further.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. A fraction consists of two parts: a numerator and a denominator. The numerator, located on top, tells us how many parts we have. The denominator, at the bottom, tells us into how many parts the whole is divided.
For example, in the fraction \(\frac{4}{5}\), 4 is the numerator and shows that four parts out of five are considered, while 5 is the denominator indicating that the total is divided into five parts. This is essential in understanding fractions as it gives us a way to compare different parts and wholes.

Finding a Common Denominator

When dealing with multiple fractions, having a common denominator makes calculations easier. This means that the denominators of both fractions become the same, allowing for straightforward addition, subtraction, and comparison.

• Find the least common multiple (LCM) of the denominators of the fractions.
• The LCM is the smallest number that both denominators can divide without leaving a remainder.
• Convert each fraction to an equivalent fraction with this common denominator.

In the example, for \(\frac{4}{5}\) and \(\frac{1}{2}\), the denominators are 5 and 2. The LCM is 10, thus we convert these fractions to \(\frac{8}{10}\) and \(\frac{5}{10}\), respectively.

Simplification of Fractions

Simplification of fractions involves reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1.

• Identify the common factors of the numerator and the denominator.
• Divide both by the greatest common factor (GCF).
• The result is a simplified fraction where the numerator and denominator are the smallest possible integers.

For instance, if the resulting fraction is not an integer, like \(\frac{40}{10}\), it is simplified to 4, because 10 divides into 40 perfectly, indicating it's already in its simplest form.

Working with Numerators

The numerator in a fraction reflects how many parts are being considered from the division defined by the denominator. When finding the least common multiple of fractions, the numerators play a crucial role.

• After converting fractions to a common denominator, focus on finding the LCM of the numerators.
• Determine any factors common to both numerators and proceed to find their product if no common factors are present.
• Example: For numerators 8 and 5 in \(\frac{8}{10}\) and \(\frac{5}{10}\), since they share no common factors aside from 1, the LCM is the product: 8 x 5 = 40.

This highlights the need to understand and compute functions involving numerators when working with fraction operations.