Problem 6 - Elementary Algebra

Question

Find the sums. \(-\frac{2}{3}+\left(-\frac{5}{3}\right)\)

Step-by-Step Solution

Answer

Answer: \(-\frac{7}{3}\)

Step 1: Identify the fractions

We are given the two fractions, \(-\frac{2}{3}\) and \(-\frac{5}{3}\). Both fractions have the same denominator (3).

Step 2: Add the numerators

Since the denominators are the same, we can directly add the numerators together: \(-2 + (-5)\). This results in a new numerator of \(-7\).

Step 3: Write the sum using the original denominator

Now, we can write the sum with the new numerator and the original denominator: \(-\frac{7}{3}\). So, the sum is \(-\frac{2}{3}+\left(-\frac{5}{3}\right) = -\frac{7}{3}\).

Key Concepts

Like Denominators

When adding fractions, a key advantage is when they have like denominators, meaning they share the same denominator. This common denominator allows us to directly add the numerators without any additional adjustments, simplifying calculations significantly.
For example, in the fractions \(-\frac{2}{3}\) and \(-\frac{5}{3}\), both fractions have a denominator of 3. Thus, they are called like fractions. This simplifies the addition process, as you don't have to find a common denominator.
Steps when dealing with like fractions:

• Ensure that both fractions have the same denominator.
• Add the numerators while keeping the denominator unchanged.

Remember, working with like denominators streamlines the process of adding fractions. It's just a matter of summing the numerators and writing them over the shared denominator.

Negative Fractions

Fractions can have negative values, either in their numerators, denominators, or both. A negative fraction such as \(-\frac{2}{3}\) signifies that the quantity or value it represents is negative. Negative fractions follow the same rules in arithmetic as positive fractions, but with an added sign consideration.
In adding fractions like \(-\frac{2}{3}\) and \(-\frac{5}{3}\), it’s important to treat the negative signs appropriately during addition. Essentially, adding two negative numbers results in a more negative number. Just like adding \-2\ and \-5\ in simpler arithmetic yields \-7\, adding \(-\frac{2}{3}\) and \(-\frac{5}{3}\) results in \(-\frac{7}{3}\) because:

• The sign of the result follows the signs of the operands, both being negative.
• The process remains the same numerically: sum the numbers as usual and keep the negative sign.

Handling negative fractions can initially seem tricky, but remembering that you add them just like any other numbers—paying heed to the signs—can make it easy.

Numerator Addition

Numerator addition is the simplest part about adding fractions with the same denominators because it involves straightforward integer addition or subtraction. Once you have established that the fractions you are adding have the same denominator, you can focus purely on their numerators.
In the case of our fractions: \-2\ and \-5\ are the numerators. When adding these two numbers, you perform the arithmetic: \-2 + -5 = -7\. This new numerator value then pairs with the common denominator, giving us our final fraction: \(-\frac{7}{3}\).
Simple steps for numerator addition:

• Add the numerators together as you would with basic integers.
• Keep track of signs: addition of negative numbers involves adding their absolute values and affixing a negative sign to the result.
• Write the result over the original, shared denominator.

With numerator addition, stay mindful of the context of negative and positive values to ensure accurate computation.