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Chapter 4: Problem 267

In the following exercises, find each sum. $$\frac{5 a}{7}+\frac{4 a}{7}$$

Short Answer

Expert verified
\(\frac{9a}{7}\)

Step by step solution

01

Identify the common denominator

Both fractions in the problem \(\frac{5a}{7}\) and \(\frac{4a}{7}\) have the same denominator, which is 7.
02

Combine the numerators

Since the denominators are the same, simply add the numerators. The numerators are 5a and 4a. So, we have: \(\frac{5a + 4a}{7}\).
03

Simplify the numerator

Combine the terms in the numerator: \5a + 4a = 9a\. This gives us \(\frac{9a}{7}\).

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Key Concepts

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

Common Denominator
Understanding the role of the common denominator is crucial when adding fractions. The common denominator is the same number on the bottom part of both fractions involved in the operation. This consistency allows you to combine the numerators (the top parts of the fractions) directly.
In our exercise, both fractions \(\frac{5a}{7}\) and \(\frac{4a}{7}\) have a denominator of 7.
When denominators are already the same, it simplifies the addition process because you can focus solely on adding the numerators.
To summarize:
  • The denominator remains the same.
  • You only need to work on the numerators.
Combining Numerators
Once you have established a common denominator, the next step is to combine the numerators. Combining the numerators means you add the numbers on top of the fractions while keeping the denominator the same.
In the exercise, we add the numerators \5a\ and \4a\. This process looks like:
\(\frac{5a}{7} + \frac{4a}{7} = \frac{5a + 4a}{7}\)
Combining these gives us:
\(5a + 4a = 9a\)
So, the fraction now becomes:\

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