Problem 57 - Elementary Algebra Within Reach

Question

In Exercises 57-60, evaluate the expression. $$ \frac{2}{3}+\frac{1}{3} $$

Step-by-Step Solution

Answer

The result is \(1\), since \(\frac{3}{3}\) equals \(1\).

Step 1: Identify the fractions

The fractions to be added are \(\frac{2}{3}\) and \(\frac{1}{3}\)

Step 2: Add the fractions

Since the denominators are same, we directly add the numerators. This gives us \(\frac{2+1}{3}\)

Step 3: Simplify the result

When you add the numerators, you get \(\frac{3}{3}\)

Key Concepts

Understanding Fractions

Fractions are a way of representing numbers that are not whole. They consist of two parts: a numerator and a denominator, with a horizontal line separating them. Think of a fraction as a way to divide something into equal parts.
For example, if you have a fraction \(\frac{2}{3}\), it represents two out of three equal parts of a whole. This can be useful when dealing with quantities that are less than one or when dividing items into smaller sections.

• Fractions show how many pieces of a whole you have.
• They are written with one number over another, separated by a line.
• The number above (numerator) tells you how many parts you have.
• The number below (denominator) tells you the total number of equal parts the whole is divided into.

Understanding fractions helps when adding them together, especially when they have the same denominator. The process becomes a simple matter of adding the numerators together.

The Role of Numerators

Numerators are the top number of a fraction. They play a crucial role because they show us how many parts of the whole are being considered. For addition of fractions, understanding numerators is key.
In our example \(\frac{2}{3} + \frac{1}{3}\), the numerators are \(2\) and \(1\). We add these numbers together when the denominators are the same. This results in the numerator \(3\) in the fraction \(\frac{3}{3}\).

• Numerator is the top number in a fraction.
• It tells you the "how many" of the parts you have.
• When adding fractions with the same denominator, add only the numerators.

Always pay attention to the numerator when determining the value of a fraction. It changes how much of the whole you have.

The Importance of Denominators

Denominators are the bottom number of a fraction. They tell us into how many equal parts the whole is divided. When adding fractions, having common denominators simplifies the process greatly.
In the expression \(\frac{2}{3} + \frac{1}{3}\), both fractions have the same denominator, which is \(3\). This means the fractions represent parts of the same whole divided into three equal parts.

• Denominator is the bottom number in a fraction.
• It shows how many total parts exist in one whole.
• Fractions can only be added directly if they have the same denominator.

Having the same denominator allows you to focus only on adding the numerators. This makes fraction addition straightforward and prevents confusion regarding the division of the whole.