Problem 38 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -\frac{5}{9}+\frac{1}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Problem

We need to add two fractions with different denominators: \(-\frac{5}{9}\) and \(\frac{1}{3}\). This means we first need to find a common denominator to perform the addition.

Step 2: Find a Common Denominator

The denominators are 9 and 3. The least common multiple of 9 and 3 is 9. Therefore, the common denominator for our fractions will be 9.

Step 3: Adjust the Fractions

Rewrite \(\frac{1}{3}\) with the common denominator. To change \(\frac{1}{3}\) to a denominator of 9, multiply both the numerator and denominator by 3: \[\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}\] Now, we have \(-\frac{5}{9}\) and \(\frac{3}{9}\).

Step 4: Add the Fractions

Now both fractions have the same denominator, so we can add them directly: \[-\frac{5}{9} + \frac{3}{9} = \frac{-5 + 3}{9} = \frac{-2}{9}\]

Step 5: Simplify the Result (if necessary)

The result \(\frac{-2}{9}\) cannot be simplified further since 2 and 9 have no common factors other than 1.

Key Concepts

Common Denominator

When dealing with fractions, a common denominator is essential for performing addition or subtraction. This process involves finding the smallest number that can be uniformly divided by both (or all) the denominators involved. For instance, when you have the fractions \(-\frac{5}{9}\) and \(\frac{1}{3}\), their denominators are 9 and 3, respectively. The smallest number that both 9 and 3 can divide is 9.

To find the common denominator:

• List the multiples of each denominator.
• Identify the smallest multiple that is common between them.
• That smallest number is your common denominator.

This step is crucial because it allows you to restructure fractions to have equivalent bases, paving the way for simple addition or subtraction. No need to fear different denominators once you master this simple trick!

Like Denominators

Once fractions share the same denominator, they are known as 'like denominators'. This means you've successfully found that common base number that both fractions can share, making addition seamless. Referring to our previous example, once \(-\frac{5}{9}\) and \(\frac{1}{3}\) are restructured so that they both have a denominator of 9 (like \(-\frac{5}{9}\) and \(\frac{3}{9}\)), it's easy to see that they are now aligned to facilitate addition.

Here's why it matters:

• Like denominators mean the denominators are identical.
• With like denominators, you can add or subtract the numerators directly.
• No additional adjustments are necessary at this stage.

Simply focus on the numerators and perform the arithmetic operation while keeping the denominator constant. With like denominators, adding fractions turns into a straightforward task.

Simplifying Fractions

Simplifying fractions is the cherry on top of your fraction operations. Once the fractions are added or subtracted, you may find that the resulting fraction can be reduced. Simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD) so that the fraction is in its simplest form.

For example, if you have the resultant fraction \(\frac{-2}{9}\), you would check if both 2 and 9 can be divided by the same number, other than 1.

• List the factors of each number.
• Find the largest shared factor.
• Divide both the numerator and denominator by this number.

In this case, since 2 and 9 have no common factors other than 1, the fraction \(\frac{-2}{9}\) is already simplified. Simplifying ensures clarity and precision in your final answer, leaving it tidy and easy to interpret.