Problem 51 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-\frac{2}{5}+\frac{1}{3}\)

Step-by-Step Solution

Answer

-\frac{1}{15}

Step 1: Find the Least Common Denominator (LCD)

Identify the denominators of the fractions: 5 and 3. Calculate the least common multiple (LCM) of these numbers to find the least common denominator. The LCM of 5 and 3 is 15.

Step 2: Convert Fractions to Equivalent Fractions with LCD

Convert \(-\frac{2}{5}\) and \(\frac{1}{3}\) to equivalent fractions with the denominator of 15. \( -\frac{2}{5} = -\frac{2 \times 3}{5 \times 3} = -\frac{6}{15} \). Similarly, \( \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \).

Step 3: Add the Converted Fractions

Now that the fractions have the same denominator, add the numerators while keeping the denominator: \( -\frac{6}{15} + \frac{5}{15} = \frac{-6 + 5}{15} = \frac{-1}{15} \).

Step 4: Simplify the Result

Verify that the fraction \( \frac{-1}{15} \) is in its simplest form. Since the numerator and the denominator have no common factors other than 1, \( \frac{-1}{15} \) is already simplified.

Key Concepts

least common denominator

When adding or subtracting fractions, it's essential that the fractions share the same denominator. The least common denominator (LCD) is the smallest number that each of the original denominators divides into evenly. For example, to add \(-\frac{2}{5}+\frac{1}{3}\), we need to find a common denominator for 5 and 3. The least common multiple (LCM) of 5 and 3 is 15, which becomes our LCD. This step ensures that our fractions are 'speaking the same language' and makes addition straightforward.

equivalent fractions

Once you find the LCD, you must convert each fraction to an equivalent fraction with this denominator. This involves multiplying both the numerator and the denominator by the same number. For example, to convert \(-\frac{2}{5}\) to a fraction with a denominator of 15, multiply numerator and denominator by 3: \(-\frac{2 \times 3}{5 \times 3} = -\frac{6}{15}\). Similarly, for \(\frac{1}{3}\), multiply both by 5: \(\frac{1 \times 5}{3 \times 5} = \frac{5}{15}\).

simplifying fractions

After adding the fractions, it's important to simplify the result. Simplifying means making the fraction as simple as possible by ensuring that the numerator and denominator share no common factors other than 1. In this case, after adding \(-\frac{6}{15} + \frac{5}{15}\), we get \(\frac{-1}{15}\). Since 1 and 15 share no common factors other than 1, \(\frac{-1}{15}\) is already in its simplest form.

negative fractions

Throughout fraction addition, handling the signs correctly is crucial. When dealing with negative fractions like \(-\frac{2}{5}\), ensure the negative sign follows through the calculation. In our example, adding \(-\frac{6}{15}\) and \(\frac{5}{15}\) involves combining \-6 + 5\, resulting in -1. This total is then placed over the common denominator of 15, giving us \(\frac{-1}{15}\).