Problem 67 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ -\frac{1}{6}-\frac{1}{6} $$

Step-by-Step Solution

Answer

-\frac{1}{3}

Step 1: Identify the operation and fractions

Notice that the problem involves adding two fractions. Both fractions are \( -\frac{1}{6} \).

Step 2: Add the fractions

Since the fractions have the same denominator, add the numerators together: \( -\frac{1}{6} + -\frac{1}{6} = -\frac{1+1}{6} \).

Step 3: Simplify the expression

Add the numerators: \( -1 + -1 = -2 \). Therefore, the sum is \( -\frac{2}{6} \).

Step 4: Simplify the fraction

Break down \( -\frac{2}{6} \) by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 2 and 6 is 2. Simplify the fraction: \( -\frac{2 \div 2}{6 \div 2} = -\frac{1}{3} \).

Key Concepts

adding fractions

When you need to add fractions, it's essential to understand a few key points. First, you should ensure that the fractions share the same denominator. For example, in the problem \( -\frac{1}{6} - \frac{1}{6} \), both fractions already have the same denominator, which is 6.
If the denominators are not the same, you have to find a common denominator. This involves finding a multiple that both denominators share. Once the denominators are the same, you proceed to add the numerators.
In our example, since the denominators are the same, we directly add the numerators:\(<-1> + < -1> = -2\). This gives us the new fraction: \( -\frac{2}{6} \).
Putting it simply, remember these steps when adding fractions:

• Ensure denominators are the same.
• Add the numerators.
• Keep the common denominator.

simplifying fractions

Simplifying fractions makes them easier to understand and work with. To simplify a fraction, you need to divide the numerator and the denominator by their greatest common divisor (GCD). This reduces the fraction to its lowest terms.
In our example, \( -\frac{2}{6} \) can be simplified. The GCD of the numerator (2) and the denominator (6) is 2. We divide both the numerator and the denominator by their GCD:\[ -\frac{2}{6} = -\frac{2 \div 2}{6 \div 2} = -\frac{1}{3} \].
Simplifying fractions ensures that they are in their most basic form, which is crucial for clarity and further calculations.

greatest common divisor

The greatest common divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator of a fraction. Understanding how to find the GCD is essential for simplifying fractions.
To find the GCD, you can use methods such as:

• Prime factorization: Breaking down both numbers into their prime factors and then choosing the highest common factors.
• Euclidean algorithm: A more systematic approach, where you repeatedly subtract the smaller number from the larger number until you get the GCD.

For instance, in the problem \( -\frac{2}{6} \), the GCD of 2 and 6 is 2. By dividing both the numerator and the denominator by their GCD, we simplify the fraction to \( -\frac{1}{3} \). Knowing how to find and use the GCD is a vital skill in working with fractions.