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Chapter 6: Problem 12

Add or subtract as indicated. $$\frac{5}{r}+\frac{9}{r}$$

Short Answer

Expert verified
\frac{14}{r}

Step by step solution

01

Identify the Common Denominator

Observe that both fractions, \(\frac{5}{r}\) and \( \frac{9}{r} \), have the same denominator, which is \( r \).
02

Combine the Numerators

Since both fractions have the same denominator, combine the numerators by adding them: \[ \frac{5}{r} + \frac{9}{r} = \frac{5 + 9}{r}. \]
03

Simplify the Result

Add the numerators: \[ 5 + 9 = 14. \] So the expression simplifies to \[ \frac{14}{r}. \]

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Key Concepts

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

common denominator
When adding fractions, the first important step is to identify a common denominator. This means that the bottom number (the denominator) in both fractions should be the same.
Why is this important? Because fractions with the same denominator represent parts of the same whole, making them easy to add or subtract.
In the example \(\frac{5}{r}\) and \(\frac{9}{r}\), both fractions already share the same denominator, \(\r\). This allows us to move on to the next step smoothly.
combine numerators
Once we have confirmed that the denominators are the same, the next step is to combine the numerators. The numerator is the top part of the fraction.
Combining the numerators involves adding them together when dealing with addition problems, or subtracting in subtraction problems.
In our example: Add the numerators \(5\) and \(9\):
\[ \frac{5}{r} + \frac{9}{r} = \frac{5 + 9}{r}. \] Combining the numerators gives us \(\frac{14}{r}\). Notice that the denominator remains the same.
simplification
The last step in solving fraction addition problems is simplification. This means making the fraction as simple as possible.
In our problem, we have already combined the fractions to form \(\frac{14}{r}\).
Since \(14\) and \(r\) do not share any common factors (other than 1), the fraction is already in its simplest form and no further simplification is needed.
Simplification often involves reducing the fraction by dividing both the numerator and the denominator by their greatest common factor, but in this case, \(14\) and \(r\) don鈥檛 need any further reduction.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{m^{2}-n^{2}}{n-m} $$

Write each rational expression in lowest terms. $$ \frac{y^{3}(y-4)}{y^{2}(y-4)} $$

Solve each problem. The number of long-distance phone calls between two cities during a certain period varies jointly as the populations of the cities, \(p_{1}\) and \(p_{2}\), and inversely as the distance between them, in miles. If 80,000 calls are made between two cities 400 mi apart, with populations of 70,000 and 100,000 , how many calls (to the nearest hundred) are made between cities with populations of 50,000 and 75,000 that are 250 mi apart?

Multiply or divide as indicated. $$ \frac{p^{2}-25}{4 p} \cdot \frac{2}{5-p} $$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{2}{3} \div \frac{8}{9} $$
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