Problem 57 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{1}{8}+\frac{3}{8} $$

Step-by-Step Solution

Answer

\( \frac{1}{8} + \frac{3}{8} = \frac{1}{2} \)

Step 1 - Check the Denominators

Identify the denominator for each fraction. In this case, both fractions have the denominator 8.

Step 2 - Add the Numerators

Since the denominators are the same, simply add the numerators. \( \frac{1 + 3}{8} \)

Step 3 - Simplify the Fraction

Add the numerators and place the result over the common denominator. \( \frac{4}{8} \)

Step 4 - Simplify Further, if Possible

Simplify \( \frac{4}{8} \) by dividing the numerator and the denominator by their greatest common divisor, which is 4. \( \frac{4 \div 4}{8 \div 4} = \frac{1}{2} \)

Step 5 - Verification

Using a calculator, verify that the sum is correct. Inputting \( \frac{1}{8} + \frac{3}{8} \) should yield 0.5, which confirms that \( \frac{1}{2} \) is the correct answer.

Key Concepts

Common Denominators

When adding or subtracting fractions, it's essential that the fractions have the same denominator, which is known as a common denominator. The denominator is the number located at the bottom of a fraction. In the problem \(\frac{1}{8} + \frac{3}{8}\), both fractions already share the same denominator of 8. This simplifies the process, as you only need to add the numerators (the numbers on top of the fraction). Having common denominators means that the parts being added together are the same size, making the operation straightforward.

Simplifying Fractions

After you've performed the addition or subtraction of fractions, it's important to simplify the resulting fraction if possible. Simplifying a fraction involves reducing it to its smallest possible form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD). In the given exercise, the fraction \(\frac{4}{8}\) can be simplified. Since the GCD of 4 and 8 is 4, you divide both the numerator and the denominator by 4. This results in \(\frac{4 \div 4}{8 \div 4} = \frac{1}{2}\), which is the simplified form. Simplifying fractions not only makes them easier to work with in further calculations but also helps in understanding and communicating the smallest possible quantity.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a key concept used to simplify fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD involves a few steps. One common method is to list out the factors of both the numerator and the denominator and then identify the highest number that appears in both lists. For example, with the fraction \(\frac{4}{8}\), the factors of 4 are 1, 2, and 4, and the factors of 8 are 1, 2, 4, and 8. The highest number common to both lists is 4, making it the GCD. After identifying the GCD, you simplify the fraction by dividing both the numerator and the denominator by this number.