Problem 47 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(\frac{-3}{5}+\frac{4}{5}\)

Step-by-Step Solution

Answer

The result is \(\frac{1}{5}\).

Step 1: Identify the denominators

Both fractions \(\frac{-3}{5}\) and \(\frac{4}{5}\) have the same denominator, which is 5.

Step 2: Add the numerators

Since the denominators are the same, add the numerators directly: \(-3 + 4 = 1\).

Step 3: Write the result as a fraction

The result of adding the numerators is the new numerator, and the common denominator remains the same. So, \(\frac{1}{5}\).

Step 4: Simplify if possible

Check if the fraction can be simplified. In this case, \(\frac{1}{5}\) is already in its simplest form.

Step 5: Check using the number line

Optionally, use the number line to verify the result: Start at \(-\frac{3}{5}\), move right by \(\frac{4}{5}\), and you will end up at \(\frac{1}{5}\).

Key Concepts

Common Denominators

When adding fractions, having common denominators is essential. The denominator is the bottom number in a fraction and it tells us into how many equal parts the whole is divided. When two fractions have the same denominator, it means we are working with parts of the same size or shape.
For example, in the exercise \(\frac{-3}{5} \) and \(\frac{4}{5} \), both fractions have a denominator of 5. This common denominator allows us to easily add the fractions together.
If the denominators were different, we would first need to find a common denominator before adding the numerators. Using a common denominator simplifies the process dramatically and ensures accuracy in combination of fractions.

Numerator Addition

Once we have established a common denominator, the next step is to add the numerators. The numerator is the top number in a fraction and it represents how many parts we have.
In our example, the numerators are \(-3\) and \(+4\). Because they have the same denominator, we can directly add them: \(-3 + 4 = 1\).
This step combines the actual parts of the fractions while maintaining the same type or size of parts, due to the common denominator. It is vital to carefully handle positive and negative signs in the numerators to get the correct result. A mistake in adding numerators will typically result in an incorrect fraction.

Simplifying Fractions

The final step in adding fractions is simplifying, if possible. Simplifying a fraction means reducing it to its simplest form. This involves ensuring there are no common factors between the numerator and the denominator other than 1.
In our exercise, we added \(\frac{-3}{5} + \frac{4}{5}\) and got \(\frac{1}{5}\). The fraction \(\frac{1}{5}\) is already in its simplest form since the numerator 1 and the denominator 5 have no common factors other than 1.
To check if a fraction can be simplified, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by this number. Simplifying ensures that our answer is as straightforward and concise as possible, which is useful for understanding and further calculations.