Problem 29 - Elementary Algebra

Question

For the following problems, perform each indicated operation. \(\frac{1}{8}+\frac{1}{2}\)

Step-by-Step Solution

Answer

Answer: The sum of the fractions \(\frac{1}{8}\) and \(\frac{1}{2}\) is \(\frac{5}{8}\).

Step 1: Find the least common multiple of the denominators

First, we need to find the least common multiple (LCM) of the denominators 8 and 2. The LCM is the smallest non-zero number that is a multiple of both denominators. Since 8 is a multiple of 2, the LCM is 8.

Step 2: Convert the fractions to equivalent fractions with the common denominator

Now that we have found the LCM to be 8, let's convert each fraction to an equivalent fraction with a denominator of 8. The first fraction is already in this form: \(\frac{1}{8}\). For the second fraction, we need to multiply both the numerator and denominator by 4 (since \(\frac{8}{2}=4\)) to get an equivalent fraction with a denominator of 8: \(\frac{1}{2} \times \frac{4}{4} = \frac{4}{8}\). So, we have: \(\frac{1}{8}+\frac{1}{2} = \frac{1}{8}+\frac{4}{8}\)

Step 3: Add the numerators and place the sum over the common denominator

Now that we have two equivalent fractions with the same denominator, we can add the numerators and place the result over the common denominator 8: \(\frac{1}{8}+\frac{4}{8} = \frac{1+4}{8}\)

Step 4: Simplify and write the answer

Finally, we will simplify the result by adding the numerators: \(\frac{1+4}{8} = \frac{5}{8}\) So, the sum of the two fractions is \(\frac{5}{8}\).

Key Concepts

Least Common Multiple

Finding the Least Common Multiple (LCM) is an essential part of fraction addition. When you need to add fractions with different denominators, the LCM comes into play by helping you find a shared foundation for the fractions. Simply put, the LCM is the smallest number that both denominators divide without leaving a remainder.
In our exercise, the fractions are \(\frac{1}{8}\) and \(\frac{1}{2}\). The denominators are 8 and 2. Since 8 is a direct multiple of 2, the LCM is 8. This means we can easily align both fractions to have the same bottom number (denominator).
To find the LCM manually, list the multiples of each denominator and identify the smallest common one:

• Multiples of 8: 8, 16, 24, ...
• Multiples of 2: 2, 4, 6, 8, 10, ...

The smallest common multiple is 8, making it the LCM necessary for the operation.

Denominator

The term "denominator" often appears when dealing with fractions. It sits at the bottom part of a fraction and signifies into how many parts the whole is divided. For example, in the fraction \(\frac{1}{8}\), the denominator is 8, meaning the whole is divided into 8 equal parts.
Different denominators complicate fraction addition because they represent divisions into different-sized parts. Therefore, the first step in adding fractions like \(\frac{1}{8}\) and \(\frac{1}{2}\) is harmonizing the denominators. Using a common denominator (the LCM) allows you to add the fractions easily.
Once the fractions share a denominator, each represents the same division of whole, paving the way for the straightforward addition of the numerators. Thus, understanding the denominator is key to mastering operations involving fractions.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same quantity or portion of a whole. They might look different, but they value the same. To convert a fraction into another equivalent one, you multiply or divide both the numerator and the denominator by the same number.
Take \(\frac{1}{2}\), for example. By multiplying the numerator (1) and the denominator (2) by 4, you get \(\frac{4}{8}\). Both fractions are equivalent because they represent the same portion of a whole.
This concept is crucial in the exercise where we converted \(\frac{1}{2}\) to \(\frac{4}{8}\) using the Least Common Multiple. By creating equivalent fractions with a common denominator, our addition becomes workable.

• The process involves finding a number that both denominators can divide (the LCM).
• Then, adjust each fraction to make their denominators identical.

Creating equivalent fractions is an essential skill in handling any arithmetic involving fractions.

Numerator

In a fraction, the numerator is the number on top. It details how many parts we have out of the whole. In \(\frac{1}{8}\), 1 is the numerator, indicating one part out of eight.Adding fractions means you eventually add their numerators once the denominators match. For \(\frac{1}{8} + \frac{4}{8}\), we add 1 and 4, getting 5. We then place this sum over the common denominator, resulting in \(\frac{5}{8}\).

• The numerator gives the quantity represented by the fraction.
• To perform fraction addition, focus on aligning denominators first.
• Then, sum up the numerators across the equivalent fractions.

Understanding and manipulating numerators is crucial to executing operations on fractions effortlessly.