Problem 27 - Elementary Algebra

Question

For exercises 27-34, evaluate. $$ \frac{1}{12}+\frac{5}{12} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Fractions

The fractions given in the problem are \(\frac{1}{12}\) and \(\frac{5}{12}\).

Step 2: Check the Denominators

Both fractions have the same denominator (12).

Step 3: Add the Numerators

Since the denominators are the same, simply add the numerators: \(1 + 5 = 6\).

Step 4: Write the Result as a Fraction

Now, write the sum of the numerators over the common denominator: \(\frac{6}{12}\).

Step 5: Simplify the Fraction

Simplify the fraction \(\frac{6}{12}\) by dividing both the numerator and the denominator by their greatest common divisor (6): \(\frac{6\div 6}{12\div 6} = \frac{1}{2}\).

Key Concepts

fraction addition

Adding fractions can seem tricky, but it's simple when we break it down into easy steps. When you add fractions, you're essentially combining parts of a whole. To start, you identify the fractions and their denominators. If the denominators are the same, as in our example \(\frac{1}{12} + \frac{5}{12}\), you just add the numerators together. The denominator remains the same. Here's a step-by-step approach:

First, identify the fractions you need to add. Next, check if the denominators are the same. If they are, move on to the next step. If not, you'll need to find a common denominator. Luckily, in our example, both denominators are already the same (12). Now, simply add the numerators (1+5 = 6). Your new fraction is \(\frac{6}{12}\).

common denominator

The common denominator is a crucial concept for adding fractions. It ensures that the parts you are adding together are of the same size. Without a common denominator, you can't simply add the numerators. For instance, if you had \(\frac{1}{4}\) and \(\frac{1}{6}\), you can't add them directly because the parts (denominators) are different.

To find a common denominator, you need the least common multiple (LCM) of the denominators. In our initial example, the denominator is already common (12). However, if the denominators were different, you would find the LCM. Suppose you need to add \(\frac{1}{4}\) and \(\frac{1}{6}\). The LCM of 4 and 6 is 12. You would adjust each fraction to have 12 as the denominator, making it easier to add. Adjust \(\frac{1}{4}\) to \(\frac{3}{12}\) and \(\frac{1}{6}\) to \(\frac{2}{12}\), then add the numerators.

simplifying fractions

Simplifying fractions is an essential step to ensure your final answer is in its simplest form. After you add fractions and get your result, like \(\frac{6}{12}\), you should reduce it if possible. Simplifying means dividing the numerator and the denominator by their greatest common divisor (GCD).

For \(\frac{6}{12}\), identify the GCD of 6 and 12, which is 6. Divide both the numerator (6) and the denominator (12) by this number. This process gives you \(\frac{6 \div 6}{12 \div 6} = \frac{1}{2}\). Hence, \(\frac{1}{2}\) is the simplified form of \(\frac{6}{12}\). Simplifying fractions not only makes them easier to understand but is also often required for final answers in math problems.