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Chapter 5: Problem 3

Find each sum or difference. Write in simplest form. \(\frac{2}{9}+\left(-\frac{4}{9}\right)\)

Short Answer

Expert verified
The sum is \(\frac{-2}{9}\).

Step by step solution

01

Identify and Understand the Problem

We need to find the sum of two fractions: \(\frac{2}{9}\) and \(-\frac{4}{9}\). Both fractions have the same denominator, which is 9.
02

Combine the Numerators

When adding or subtracting fractions with the same denominator, we keep the denominator unchanged and perform the operation on the numerators. So, we add the numerators as follows: \(2 + (-4) = 2 - 4\).
03

Simplify the Numerator

Calculate the result from the previous step: \(2 - 4 = -2\). Therefore, the sum of the numerators is \(-2\).
04

Write the Resulting Fraction

Combine the result of the numerators with the common denominator: \(\frac{-2}{9}\). This fraction is already in its simplest form since the greatest common divisor of 2 and 9 is 1.

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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 a fundamental concept in mathematics. When you add fractions, the key is to have a common denominator. This makes it easy to combine the numerators effectively. In our example, both fractions \(\frac{2}{9}\) and \(-\frac{4}{9}\) have the same denominator, 9, which simplifies the process.
  • Keep the common denominator the same.
  • Add the numerators together: \(2 + (-4)\).
Always remember that when dealing with negative numbers in addition, the operation translates to subtraction. Here, that means performing \(2 - 4 = -2\). Thus, the resulting fraction after adding is \(\frac{-2}{9}\).
Subtracting Fractions
Subtracting fractions works very similarly to adding them. However, sometimes subtraction can be less intuitive because it inherently involves the concept of taking away.
  • Ensure the denominators are the same.
  • Subtract the numerators: \(2 - 4\).
In this exercise, we're effectively subtracting \(\frac{4}{9}\) from \(\frac{2}{9}\), which results in a negative number because 4 is greater than 2. When you have a negative result for the numerator, simply write the fraction as \(\frac{-2}{9}\), maintaining the common denominator of 9.
Simplifying Fractions
Simplifying fractions is about making the fraction as simple as possible. This often involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
  • Check if the numerator and denominator have common factors.
  • If they share a common factor, divide both by this number.
In our solution, the fraction \(\frac{-2}{9}\) is already simplified. There's no number other than 1 that divides both 2 and 9 evenly. Thus, \(\frac{-2}{9}\) remains as it is. Simplifying fractions ensures that they are written in the most compact form possible.

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Most popular questions from this chapter

Estimate each product. $$-5 \frac{1}{3} \cdot 3 \frac{4}{5}$$

Solve each equation. Check your solution. $$\frac{1}{3} n=\frac{2}{9}$$

Estimate each product. $$2 \frac{1}{4} \cdot 2 \frac{1}{9}$$

Use exponents to write the prime factorization of each number or monomial. 60

Find the GCF of each pair of monomials. $$4 p^{2}, 6 p$$
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