Problem 17 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(-\frac{3}{4}+\left(-\frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The sum is \(-\frac{3}{2}\).

Step 1: Identify the operation

We need to find the sum of two fractions: \(-\frac{3}{4}\) and \(-\frac{3}{4}\). This translates to adding these fractions together.

Step 2: Check the denominators

Both fractions have the same denominator which is 4. This allows us to combine the numerators directly without finding a common denominator.

Step 3: Add the numerators

Since the denominators are the same, we add the numerators:\(-3 + (-3) = -6\).Thus, our combined fraction is \(\frac{-6}{4}\).

Step 4: Simplify the fraction

To simplify \(\frac{-6}{4}\), divide both the numerator and the denominator by their greatest common divisor, which is 2. This gives us \(\frac{-3}{2}\) as the fraction in simplest form.

Key Concepts

Fractions

Fractions represent parts of a whole. They're made up of a numerator and a denominator. The numerator tells us how many parts we have, while the denominator tells us into how many parts the whole is divided. For example, in the fraction \(rac{3}{4}\), 3 is the numerator, and 4 is the denominator. This means we have 3 parts out of a total of 4 parts.

Fractions can be positive or negative, just like whole numbers. A negative fraction, like \(-\frac{3}{4}\), simply means it's on the negative side of the number line. Understanding fractions is essential because they appear frequently in math and real-life situations, like cooking or dividing a pizza among friends.

Addition of Fractions

When adding fractions, you must consider the denominators. If the denominators are the same, you can add the numerators directly. This is the case in our exercise with fractions \(-\frac{3}{4}\) and \(-\frac{3}{4}\). Both have a denominator of 4, so you can simply add the numerators:

• Add the numerators: \(-3 + (-3) = -6\).
• Keep the denominator the same: 4.

This gives you a new fraction: \(-\frac{6}{4}\). It's essential to keep the denominator the same to accurately represent the amount of parts you're working with.

When working with fractions with different denominators, you need to find a common denominator before adding, which requires additional steps. But in cases like this, with identical denominators, the process is straightforward.

Simplifying Fractions

Once you have your new fraction, it's important to simplify it. Simplifying fractions means reducing the fraction to its simplest form, where the numerator and denominator have no common factors except 1. To simplify \(\frac{-6}{4}\), find the greatest common divisor (GCD) of both the numerator and denominator.

In this case, the GCD is 2:

• Divide the numerator: \(-6 \div 2 = -3\).
• Divide the denominator: \(4 \div 2 = 2\).

This results in the simplified fraction \(\frac{-3}{2}\).

Always ensure fractions are expressed in their simplest form, as it's the easiest way to interpret and work with them effectively. Simplifying fractions also makes calculations easier and minimizes mistakes in future operations.