Problem 22 - Intermediate Algebra

Question

Perform the operations. See Example 1 . $$ -\frac{3}{4}+\left(-\frac{1}{5}\right) $$

Step-by-Step Solution

Answer

The result of the operation is \(-\frac{19}{20}\).

Step 1: Identify the Operation

We're given the expression \(-\frac{3}{4} + \left(-\frac{1}{5}\right)\). We need to perform the operation of adding two fractions.

Step 2: Common Denominator

To add the fractions, we first find a common denominator. The denominators are 4 and 5. The least common multiple of 4 and 5 is 20. Therefore, the common denominator is 20.

Step 3: Convert to Equivalent Fractions

Convert each fraction to an equivalent fraction with a denominator of 20. For \(-\frac{3}{4}\), multiply both the numerator and the denominator by 5:\[-\frac{3}{4} = -\frac{3 \times 5}{4 \times 5} = -\frac{15}{20}\]For \(-\frac{1}{5}\), multiply both the numerator and the denominator by 4:\[-\frac{1}{5} = -\frac{1 \times 4}{5 \times 4} = -\frac{4}{20}\]

Step 4: Add the Fractions

Now that both fractions have a common denominator, we add the numerators:\[-\frac{15}{20} + -\frac{4}{20} = \frac{-15 + (-4)}{20} = -\frac{19}{20}\]

Step 5: Simplify the Result

The fraction \(-\frac{19}{20}\) is already in its simplest form since 19 is a prime number and does not share any factors with 20.

Key Concepts

Common Denominator

When adding fractions, having a common denominator is crucial. A common denominator means both fractions share the same base number in the denominator, which allows the numerators to be added directly. The process begins by identifying a shared denominator that both fractions can convert into. In our example with fractions \(-\frac{3}{4}\) and \(-\frac{1}{5}\), the denominators 4 and 5 need to be unified, so we find a single number they can both divide evenly into.
By determining this denominator, we can convert each fraction without changing its value, which simplifies the process of addition. In general, using a common denominator is the backbone of fraction addition. It not only makes the arithmetic simpler but ensures accuracy. Without it, adding fractions directly would yield incorrect results.

Least Common Multiple

The least common multiple (LCM) is essential in finding that shared denominator for the fractions. It is the smallest number, that is a multiple of both denominators. To find the LCM, list the multiples of each denominator and identify the smallest common value.
Let's illustrate this with our denominators of 4 and 5:

• Multiples of 4: 4, 8, 12, 16, 20, 24, ...
• Multiples of 5: 5, 10, 15, 20, 25, ...

From the lists above, 20 is the first common multiple, making it the least common multiple. This LCM then becomes the common denominator. Gaining confidence in working with LCMs streamlines adding not just two, but potentially multiple fractions with different denominators. Understanding and applying LCM ensures a smooth transition from individual fractions to a precise combined answer.

Equivalent Fractions

Once the common denominator is established, the next step is converting fractions to equivalent forms. This means adjusting the fractions so they match the new denominator but maintain the same value. To achieve this, multiply both the numerator and the denominator by the same number.
Using our example fractions:

• For \(-\frac{3}{4}\), multiply both 3 and 4 by 5 to get \(-\frac{15}{20}\).
• For \(-\frac{1}{5}\), multiply both 1 and 5 by 4 to get \(-\frac{4}{20}\).

By doing this, each fraction is rewritten with a denominator of 20, making it possible to add them easily. Despite the transformation, the fractions’ values remain unchanged. Mastery of creating equivalent fractions enhances flexibility and ability in working with varied math problems. This step is pivotal in ensuring each fraction is ready for addition or subtraction.