Problem 25 - Beginning and Intermediate Algebra

Question

Find each sum. $$ -\frac{1}{6}+\frac{2}{3} $$

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1: Find a common denominator

Identify the denominators of the fractions involved. The denominators are 6 and 3. The least common denominator (LCD) of 6 and 3 is 6.

Step 2: Adjust fractions to have the common denominator

Convert each fraction to have the denominator of 6. The first fraction \(-\frac{1}{6}\) already has this denominator. Convert \(\frac{2}{3}\) to a fraction with denominator 6: \(\frac{2}{3} = \frac{2 \cdot 2}{3 \cdot 2} = \frac{4}{6}\).

Step 3: Add the fractions

Add the fractions with the common denominator: \(-\frac{1}{6} + \frac{4}{6} = \frac{-1 + 4}{6} = \frac{3}{6}\).

Step 4: Simplify the fraction

Simplify \(\frac{3}{6}\) by dividing the numerator and denominator by their greatest common divisor (GCD), which is 3: \(\frac{3 \div 3}{6 \div 3} = \frac{1}{2}\).

Key Concepts

Common Denominator

When adding or subtracting fractions, it's essential to ensure that the fractions share the same denominator. This common denominator allows you to combine the fractions easily.
Start by identifying the denominators of the fractions involved. For example, in the exercise \(-\frac{1}{6}+\frac{2}{3}\), the denominators are 6 and 3.

We need to find a number that both denominators can divide into evenly. This number is called the Least Common Denominator (LCD). In our case, that number is 6, as both 6 and 3 can divide evenly into 6.
Once we have the common denominator, we can convert the fractions, if necessary, so both have it. This makes the addition or subtraction straightforward.
For instance,

• The fraction \(-\frac{1}{6}\) already has the common denominator of 6.
• For \frac{2}{3}\, we convert it to a fraction with denominator 6: \frac{2}{3} = \frac{2 \cdot 2}{3 \cdot 2} = \frac{4}{6}\.

With both fractions having the common denominator, adding or subtracting them is straightforward.

Least Common Denominator

Finding the least common denominator (LCD) is a crucial step when working with fractions. The LCD is the smallest number that is a multiple of both denominators.
To find the LCD, list the multiples of each denominator and identify the smallest multiple they share.
For denominators 6 and 3, list their multiples:

• Multiples of 6: 6, 12, 18, ...
• Multiples of 3: 3, 6, 9, 12, ...

Here, 6 is the smallest common multiple. Therefore, the LCD of 6 and 3 is 6. With the LCD identified, you can convert other fractions to have this common denominator, making addition or subtraction simpler and more manageable.

Simplification

Once you have performed the addition or subtraction of fractions, the next step is to simplify the resulting fraction.
Simplification involves reducing the fraction to its smallest possible form. To do this, you need to identify the greatest common divisor (GCD) of the numerator and the denominator.
In our example, after adding \-\frac{1}{6}+\frac{4}{6}\, we get \frac{3}{6}\.
The GCD of 3 and 6 is 3 and simplifying the fraction involves dividing the numerator and the denominator by this GCD:\[\frac{3} {6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}\.\]
So, the simplified result is \frac{1}{2}\.
Simplifying fractions is important because it makes them easier to work with and understand.