Problem 40 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -\frac{5}{6}+\left(-\frac{2}{3}\right) $$

Step-by-Step Solution

Answer

The sum is \(-\frac{3}{2}\).

Step 1: Rewrite the Problem

The given problem is to add \(-\frac{5}{6}\) and \(-\frac{2}{3}\). We need to rewrite it as a single addition expression: \(-\frac{5}{6} + \left(-\frac{2}{3}\right)\). We can rewrite this as \(-\frac{5}{6} - \frac{2}{3}\).

Step 2: Find a Common Denominator

To add the fractions, we'll convert them to have a common denominator. The denominators are 6 and 3. The least common denominator (LCD) of 6 and 3 is 6.

Step 3: Convert to Equivalent Fractions

The fraction \(-\frac{5}{6}\) already has the denominator 6. Convert \(-\frac{2}{3}\) to have the same denominator by multiplying both the numerator and the denominator by 2: \(-\frac{2}{3} = -\frac{4}{6}\).

Step 4: Add the Fractions

Now add the fractions \(-\frac{5}{6}\) and \(-\frac{4}{6}\). Combine their numerators: \(-5 + (-4) = -9\). So, the result is \(-\frac{9}{6}\).

Step 5: Simplify the Fraction

Simplify \(-\frac{9}{6}\) by dividing both the numerator and denominator by their greatest common divisor, which is 3. So we get \(-\frac{9 \div 3}{6\div 3} = -\frac{3}{2}\).

Key Concepts

Common Denominator

When adding fractions, one of the most crucial steps is to find a common denominator. A common denominator is a shared multiple of the denominators of the fractions you are working with. This is essential because you can only directly add fractions when they have the same denominator.
To find a common denominator, especially the least common denominator (LCD), you need to identify the smallest multiple that both denominators share. In our example with the fractions \(-\frac{5}{6}\) and \(-\frac{2}{3}\), the denominators are 6 and 3.
To find their LCD:

• List the multiples of each denominator.
• The multiples of 6 are: 6, 12, 18, etc.
• The multiples of 3 are: 3, 6, 9, 12, etc.
• The smallest multiple they share is 6.

Thus, 6 becomes the common denominator, allowing both fractions to be compared and added easily.

Equivalent Fractions

Once you have a common denominator, the next step is to convert each fraction to an equivalent fraction with that denominator. Equivalent fractions represent the same portion of a whole, just expressed differently.
In the exercise, we need to convert \(-\frac{2}{3}\) to a fraction with a denominator of 6. To do this, you multiply both the numerator and the denominator by the same number that makes the denominators match:

• For \(-\frac{2}{3}\), multiply the numerator and the denominator by 2.
• This gives us \(-\frac{4}{6}\), which is equivalent to \(-\frac{2}{3}\).

Achieving equivalent fractions ensures that the fractions can be added or subtracted seamlessly, since they now share a common denominator.

Simplifying Fractions

After adding the fractions, you will often end up with a fraction that can be simplified to its lowest terms. Simplifying a fraction means making the fraction as simple as possible by ensuring the numerator and the denominator are as small as possible while still maintaining the same ratio.
In the example, we arrived at \(-\frac{9}{6}\) after adding the fractions. To simplify this:

• Find the greatest common divisor (GCD) of the numerator and the denominator. Here, it is 3.
• Divide both the numerator and the denominator by the GCD (3): \(-\frac{9}{3} = -3\) and \(\frac{6}{3} = 2\).
• The simplified fraction is \(-\frac{3}{2}\).

Simplifying fractions helps in understanding and using the result more efficiently, as it presents the fraction in its simplest and most understandable form.