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Chapter 4: Problem 9

Add and simplify. $$ \frac{2}{9 x}+\frac{4}{9 x} $$

Short Answer

Expert verified
\(\frac{2}{3x}\)

Step by step solution

01

Identify Common Denominator

Both fractions have the same denominator, which is \(9x\). Since the denominators are identical, we can add the numerators directly.
02

Add the Numerators

Add the numerators of the fractions: \(2+4 = 6\). The common denominator remains \(9x\). So, the sum of the fractions is \(\frac{6}{9x}\).
03

Simplify the Fraction

Notice that both the numerator and the denominator of \(\frac{6}{9x}\) can be simplified by dividing them by 3. This gives us \(\frac{2}{3x}\) after simplification.

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Key Concepts

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

Adding Fractions
Adding fractions is simpler than it might initially seem, especially when both fractions have the same denominator. When fractions share a common denominator, we can directly combine their numerators while keeping the denominator unchanged. In the given example, \(\frac{2}{9x}+\frac{4}{9x}\), both fractions have the same denominator, \(9x\). This means:
  • The denominators "line up," allowing easy addition.
  • We add only the numerators: \(2 + 4 = 6\).
Keeping the denominator constant, the result is \(\frac{6}{9x}\). It's just like adding whole numbers, except the denominator remains fixed throughout the process.
Common Denominator
A common denominator is crucial when adding fractions because it aligns the fractions on the same base, enabling direct addition or subtraction. In situations where fractions don't initially share a denominator, you would need to find the least common denominator (LCD), which is the smallest multiple common to all denominators.However, in the provided exercise, both fractions already share a common denominator \(9x\). This simplifies our work, as we don't need to adjust or find another denominator. In more complex problems where fractions do not share denominators, you might:
  • Calculate the least common multiple (LCM) of the denominators.
  • Adjust each fraction by multiplying the numerator and denominator by the necessary factor to achieve the LCM.
For now, since \(9x\) is shared by both, we were able to directly add the numerators in the simplest way.
Numerator and Denominator Simplification
Simplifying fractions is a way to express them in their smallest form without changing their value. The simplification process often involves dividing both the numerator and the denominator by their greatest common divisor (GCD).In our example, after adding the fractions, we had \(\frac{6}{9x}\). The numerator 6 and the term in the denominator 9x share a common factor, which is 3. Thus, we:
  • Divide both 6 in the numerator and 9x in the denominator by 3.
  • This gives us \(\frac{6 梅 3}{9x 梅 3} = \frac{2}{3x}\).
The result \(\frac{2}{3x}\) is the simplified form of the fraction, ensuring clarity and ease of interpretation. Reducing fractions to their simplest form makes them easier to understand and use in further calculations.

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Most popular questions from this chapter

Write each mixed number as an improper fraction. See Example 20. $$114 \frac{2}{7}$$

The area of the rectangle below is 12 square meters. If its width is \(2 \frac{4}{7}\) meters, find its length.

Solve. Which of the following are equivalent to \(7 \frac{3}{4} ?\) a. \(6 \frac{7}{4}\) b. \(5 \frac{11}{4}\) c. \(7 \frac{12}{16}\) d. all of them

$$\text { Evaluate each expression if } x=\frac{3}{4} \text { and } y=-\frac{4}{7}$$ $$\frac{2+x}{y}$$

Recall that to find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of \(\frac{1}{2}\) and \(\frac{3}{4}\) is \(\frac{\frac{1}{2}+\frac{3}{4}}{2}\). Find the average of each pair of numbers. Study Exercise \(67 .\) Without calculating, \(\operatorname{can} \frac{1}{3}\) be the average of \(\frac{1}{2}\) and \(\frac{8}{9}\) ? Explain why or why not.
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