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Chapter 6: Problem 6

Perform the indicated operations and simplify. $$\frac{2}{13}+\frac{6}{13}$$

Short Answer

Expert verified
The simplified result is \( \frac{8}{13} \).

Step by step solution

01

Identify Similar Denominators

Both fractions in the expression have the same denominator, 13. This allows us to add them directly.
02

Add the Numerators

Since the denominators are the same, we only need to add the numerators. Add 2 and 6: \[ 2 + 6 = 8 \]
03

Write the Resulting Fraction

The added sum of the numerators forms the new numerator with the common denominator, 13. Thus, the resulting fraction is: \[ \frac{8}{13} \]
04

Simplify the Fraction

Determine if the fraction \( \frac{8}{13} \) can be simplified. Since 8 and 13 have no common factors other than 1, the fraction is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When adding or subtracting fractions, it is essential to have a ***common denominator***. The denominator is the number below the fraction line, indicating into how many parts the whole is divided. In this exercise, both fractions have a denominator of 13, which is already common.

By having the same denominator, the fractions represent quantities split into the same number of parts, making the addition straightforward. You simply add the numerators, which are above the line, while maintaining the same denominator.

To sum up:
  • A common denominator allows for direct addition of numerators.
  • Both fractions here already have the common denominator of 13.
  • No need to adjust the fractions further before adding.
Simplifying Fractions
***Simplifying fractions*** involves reducing a fraction to its simplest form. This means the numerator and the denominator have no common factors apart from 1. It's an important step that helps make the fraction easier to understand and work with.

After performing an operation, like addition, always check if the resulting fraction can be simplified.

For our fraction, \( \frac{8}{13} \), observe these points:
  • 8 and 13 do not share any common factors except for 1.
  • The fraction is already in its simplest form, as we can't divide both parts by the same number greater than 1.
  • Simplification ensures clarity and accuracy in fraction operations.
Numerators and Denominators
Each fraction is made up of two parts: the ***numerator*** and ***denominator***.

The numerator is the number above the line, representing how many parts we are considering. In \( \frac{2}{13} \), the numerator is 2, and in \( \frac{6}{13} \), it is 6.

The denominator, on the other hand, is the number below the line, telling us into how many equal parts the whole is divided. Here, the denominator for both fractions is 13. This uniform denominator allows for the direct addition of numerators.

Remember these essentials:
  • Add numerators only after confirming common denominators.
  • The denominator in addition stays the same; it does not change.
  • Each part of the fraction has a distinct and important role.
Understanding the roles of numerators and denominators is key to mastering fraction addition.

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