Problem 91 - Elementary Algebra

Question

For exercises 81-96, evaluate. $$ -\frac{8}{9}+\frac{2}{3} $$

Step-by-Step Solution

Answer

- \(\frac{2}{9}\)

Step 1: Find a Common Denominator

To add the fractions \(- \frac{8}{9} + \frac{2}{3} \), first find a common denominator. The denominators are 9 and 3. The least common multiple of 9 and 3 is 9.

Step 2: Convert Fractions to Common Denominator

Rewrite \(- \frac{8}{9} \) as it is since it already has the denominator of 9. Rewrite \(\frac{2}{3} \) as \(\frac{2 \times 3}{3 \times 3} = \frac{6}{9} \) so that it has the same denominator.

Step 3: Combine the Fractions

Now that both fractions have the same denominator, combine them: \(- \frac{8}{9} + \frac{6}{9} \).

Step 4: Add the Numerators

Add the numerators: \(-8 + 6 = -2 \). Therefore, \(- \frac{8}{9} + \frac{6}{9} = - \frac{2}{9} \).

Key Concepts

common denominator

To add fractions effectively, it's crucial to understand the idea of a common denominator. The denominator is the number below the line in a fraction, indicating into how many equal parts the unit is divided.
To find a common denominator, look for the least common multiple (LCM) of both denominators involved. In the exercise \(-\frac{8}{9} + \frac{2}{3}\), denominators are 9 and 3. The LCM of 9 and 3 is 9, making 9 the common denominator.
By finding a common denominator, you make the fractions 'like', which allows for straightforward addition or subtraction.

like fractions

Like fractions are fractions that have the same denominator. In our example, we convert fractions to have the same denominator to simplify the addition process.
\( \frac{2}{3} \) can be converted by multiplying both the numerator and the denominator by 3, resulting in \( \frac{6}{9} \). Now the fractions are \( -\frac{8}{9} \) and \( \frac{6}{9} \).
Having like fractions enables you to focus only on the numerators during addition or subtraction, which simplifies calculations.

numerator addition

Once fractions have the same denominator, you can add or subtract the numerators directly. Remember, the denominator remains unchanged.
In our exercise, we add the numerators of \( -\frac{8}{9} \) and \( \frac{6}{9} \), leading to \( -8 + 6 = -2 \).
This gives us \( -\frac{2}{9} \). Adding numerators is a straightforward step once the fractions are like, simplifying the arithmetic further.

simplifying fractions

After you've combined fractions, always check if they can be simplified. A fraction is simplified when the numerator and denominator have no common factors other than 1.
In the result \( -\frac{2}{9} \), 2 and 9 share no common factors (except 1), so it is already in its simplest form.
Simplifying fractions can sometimes help in further calculations, making the fraction easier to understand and work with.