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Chapter 4: Problem 275

In the following exercises, find each sum. $$-\frac{9}{19}+\frac{17}{19}$$

Short Answer

Expert verified
\(\frac{8}{19}\)

Step by step solution

01

Identify the common denominator

Both fractions \(-\frac{9}{19}\) and \(\frac{17}{19}\) already have the same denominator, which is 19.
02

Add the numerators

Since the denominators are the same, add the numerators: \(-9 + 17\).
03

Calculate the sum of the numerators

Calculate \(-9 + 17 = 8\).
04

Write the fraction with the sum of the numerators

Place the sum of the numerators over the common denominator: \(\frac{8}{19}\).

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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, finding a common denominator is crucial. The common denominator is the same bottom number in both fractions. It acts as a universal size comparison for the parts of the fractions. In this exercise, \(-\frac{9}{19} + \frac{17}{19}\), both fractions already share \(19\) as the common denominator. This makes the addition simpler. Whenever fractions have different denominators, convert them to share a common one. This helps avoid mistakes in calculations.
Numerator
The numerator is the top part of a fraction, representing how many parts of the whole we are dealing with. For instance, in \(-\frac{9}{19}\), the \(-9\) is the numerator. In the context of this exercise, adding fractions involves only manipulating the numerators: \(-9+17\). Keep the denominator stationary during this process. The correct numerator is crucial for accurate fraction addition.
Fraction Addition
Adding fractions can seem challenging, but it follows straightforward rules. First, ensure both fractions share the same denominator. For our exercise, they already do: \(19\). Next, add the numerators while maintaining the common denominator. In this case: \(-9 + 17 = 8\). Finally, write the sum over the common denominator: \frac{8}{19}\. Remember, only change the numerators during the addition process, not the denominator.
Negative Fractions
Negative fractions might look tricky, but they follow the same principles as positive fractions. The only difference is the negative sign in the numerator, which indicates a value less than zero. When adding a negative fraction like \(-\frac{9}{19}\), treat the negative sign as any normal subtraction. For instance, \(-9 + 17\) is the same as \(17 - 9\), yielding \(8\). Understanding the role of negative numerators simplifies the addition process.

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Most popular questions from this chapter

In the following exercises, simplify. $$\frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}}$$

In the following exercises, subtract. $$\frac{3}{2}-\left(\frac{3}{2}\right)$$

In the following exercises, simplify. $$ \frac{18 r}{27 s} $$

In the following exercises, change to equivalent fractions using the given LCD. $$\frac{3}{8} \text { and } \frac{5}{6}, \quad L C D=24$$

In the following exercises, simplify. $$ \frac{9 a}{10} \div \frac{15 a}{8} $$
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