Problem 6 - Fundamentals of Mathematics

Question

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

Step-by-Step Solution

Answer

The sum is \( -\frac{7}{3} \).

Step 1: Identify the Terms

We have two terms to sum: \(-\frac{2}{3}\) and \(-\frac{5}{3}\). Both terms are negative fractions with the same denominator.

Step 2: Add the Numerators

Since the denominators of both fractions are the same, we add the numerators. Adding the numerators gives us:\(-2 + (-5) = -7\).

Step 3: Keep the Denominator

The denominator remains the same because it is common for both fractions. Thus, we have the sum as:\[ \frac{-7}{3} \].

Key Concepts

Understanding Negative Fractions

Negative fractions might seem tricky at first, but they follow the same basic rules as positive fractions. The only difference is that they have a minus sign.

Remember, fractions consist of a numerator (top number) and a denominator (bottom number). A negative sign in front affects the whole fraction. For example, \(-\frac{2}{3}\) means that the value is below zero, just like negative whole numbers.

• Negative fractions can be visualized on a number line, lying to the left of zero.
• Treat the negative sign like you would in regular subtraction or addition.
• If both fractions in an operation are negative, the result becomes more negative.

Understanding this concept is key to solving problems involving negative fractions.

Finding Common Denominators

A common denominator is essential when adding, subtracting, or comparing fractions. It ensures that the fractions are in terms that can be directly combined or compared.

• A common denominator is typically the same as the least common multiple (LCM) of the denominators involved.
• With a common denominator, you can focus on adding or subtracting just the numerators.

In our earlier exercise, the fractions \(-\frac{2}{3}\) and \(-\frac{5}{3}\) already share a common denominator of 3, which simplifies the process considerably. You don’t need to adjust the denominators, just work with what's given. This makes combining them straightforward since you only need to perform operations on the numerators.

The Process of Adding Numerators

Adding numerators is straightforward, especially when you have a common denominator. You simply focus on the numbers at the top of the fractions.

In our example, with \(-\frac{2}{3}\) and \(-\frac{5}{3}\), an important detail is that both numerators are negative. This means:

• Think of it as adding two negative numbers: \-2 + (-5) = -7\.
• The negative sign means you're moving further away from zero on the number line.

Once the numerators are summed, your final fraction keeps the same denominator, which is 3 in this situation.

So, the result is \(-\frac{7}{3}\). Remember, only the top part of the fraction changes during addition; the bottom remains unchanged with the common denominator.