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Chapter 6: Problem 10

Perform the indicated operations. Simplify when possible. $$ \frac{2}{5 n}+\frac{8}{5 n} $$

Short Answer

Expert verified
The simplified form is \(\frac{2}{n}\).

Step by step solution

01

Identify Like Terms

Notice that both fractions \(\frac{2}{5n}\) and \(\frac{8}{5n}\) have the same denominator, which is \(5n\). This makes them like terms and allows us to add the numerators directly.
02

Add Numerators

Since the denominators are the same, add the numerators together: \(2 + 8\). This gives us \(10\).
03

Write the Resulting Fraction

Place the sum of the numerators over the common denominator: \(\frac{10}{5n}\).
04

Simplify the Fraction

Simplify \(\frac{10}{5n}\). Divide the numerator and denominator by their greatest common divisor, which is \(5\). This gives us \(\frac{10 \div 5}{5n \div 5} = \frac{2}{n}\).

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Key Concepts

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

Common Denominators
When adding fractions, one must ensure that the fractions have the same denominator, known as the common denominator. In the problem \(\frac{2}{5n}+\frac{8}{5n}\), both fractions already have the same denominator, which is terrific since it simplifies the process. A common denominator allows us to add the fractions easily by focusing solely on the numerators. Think of the common denominator as a shared base that the fractions stand on. Without this shared base, direct addition isn't possible, so we would need to find one through multiplication or another method.
Simplifying Fractions
Simplifying fractions makes the results easier to understand and work with. After performing operations like addition, you may end up with a fraction that can be reduced to a simpler form. In our problem, after adding the fractions, we get \(\frac{10}{5n}\). While it鈥檚 correct, it鈥檚 not in its simplest form. To simplify it, we divide both the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD is 5. So, \(\frac{10 \, \text{div} \, 5}{5n \, \text{div} \, 5} = \frac{2}{n}\). Simplification makes the fraction cleaner, which is why it鈥檚 a standard practice.
Like Terms
Like terms are terms that have identical denominators or variables and can be combined directly. In our given fractions, \(\frac{2}{5n}\) and \(\frac{8}{5n}\) are like terms because they share the same denominator \(\frac{1}{5n}\). This similarity allows us to add or subtract them without further adjustments to their denominators. Always look to identify like terms when working with fractions, as it can significantly simplify your calculations and save time.

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