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Chapter 7: Problem 9

Add or subtract. Write answer in lowest terms. \(\frac{4}{m}+\frac{7}{m}\)

Short Answer

Expert verified
\(\frac{11}{m}\)

Step by step solution

01

Identify the common denominator

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

Add the numerators

Since the denominators are the same, add the numerators directly: \(4 + 7 = 11\).
03

Write the fraction

Place the sum of the numerators over the common denominator: \(\frac{11}{m}\).
04

Simplify the fraction

Check if the fraction \(\frac{11}{m}\) can be simplified further. Since 11 is a prime number and does not share any common factors with \(m\), the fraction is already in its lowest terms.

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Key Concepts

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

Common Denominator
A common denominator is the same denominator for all fractions involved in an operation. It is a crucial concept for adding or subtracting fractions. This is because fractions need to have matching denominators to be combined.

In our exercise, both fractions \(\frac{4}{m}\) and \(\frac{7}{m}\) share the common denominator \(m\).

When dealing with more complex fractions, you may need to find a common denominator by determining the least common multiple (LCM) of the denominators. This ensures the fractions are equivalent and can be easily added or subtracted.
Adding Fractions
To add fractions, you need to sum their numerators while keeping the common denominator the same. If the fractions don't initially have a common denominator, you'll first need to convert them so they do.

For example, adding \(\frac{4}{m}\) and \(\frac{7}{m}\) is straightforward because they already have the common denominator \(m\). Just add the numerators: \(4 + 7 = 11\).

The result is then written as a single fraction: \(\frac{11}{m}\).

When you have different denominators, you'll first find a common denominator, convert the fractions, and then add the numerators.
Simplifying Fractions
Simplifying fractions involves reducing them to their lowest terms. This means making the numerator and denominator as small as possible, without changing the value of the fraction.

To simplify a fraction, look for the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number. In our exercise, \(\frac{11}{m}\) is already simplified because 11 is a prime number and has no common factors with \(m\).

Remember, a fraction is fully simplified when you can't divide the numerator and denominator by any number other than 1 without ending up with a fraction equivalent to the original.

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Most popular questions from this chapter

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{2}{3}+\frac{8}{27} $$

Solve each problem involving direct or inverse variation. If \(m\) varies inversely as \(p^{2},\) and \(m=20\) when \(p=2,\) find \(m\) when \(p=5\)

The distance from Seattle, Washington, to Victoria, British Columbia, is about 148 mi by ferry. It takes about 4 hr less to travel by the same ferry from Victoria to Vancouver, British Columbia, a distance of about 74 mi. What is the average rate of the ferry?

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{m^{2}+3 m+2}{m^{2}+5 m+4} \cdot \frac{m^{2}+10 m+24}{m^{2}+5 m+6}$$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{m^{2}+8 m n+7 n^{2}}{m^{2}+m n-42 n^{2}} \div \frac{m^{2}-3 m n-4 n^{2}}{m^{2}-m n-30 n^{2}}$$
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