Problem 64 - Elementary and Intermediate Algebra

Question

Add. $$ -\frac{3}{4}+\frac{1}{4} $$

Step-by-Step Solution

Answer

The sum is \(-\frac{1}{2}\).

Step 1: Understand the Problem

You are tasked with adding two fractions: \(-\frac{3}{4}\) and \(\frac{1}{4}\). Both fractions have the same denominator, which makes it straightforward to combine them.

Step 2: Add the Fractions

Since the fractions \(-\frac{3}{4}\) and \(\frac{1}{4}\) have the same denominator, you can add them by adding their numerators: \(-3 + 1\). This gives you \(-2\).

Step 3: Write the Sum as a Single Fraction

The result of adding the numerators is \(-2\), and the common denominator is \(4\). So, the sum of the fractions is \(-\frac{2}{4}\).

Step 4: Simplify the Fraction

You can simplify \(-\frac{2}{4}\) by dividing both the numerator and the denominator by their greatest common divisor, which is \(2\). This gives you \(-\frac{1}{2}\).

Key Concepts

Understanding the Common Denominator

A common denominator is essential when adding or subtracting fractions. It is the shared value in the denominator position that allows direct comparison and combination of the fraction values. In our exercise, both fractions \(-\frac{3}{4}\) and \(\frac{1}{4}\) already share the denominator \(4\).
This makes the task much simpler, as we don't need to make any adjustments before adding them.
When fractions have the same denominator, you only need to add or subtract the numerators, keeping the denominator constant. This method lets you quickly perform operations without complex steps.
In cases where fractions do not share a common denominator, you'd first need to convert each fraction into an equivalent one with a common denominator. This is done through finding the least common multiple of the denominators involved.

Steps for Simplifying Fractions

Simplifying fractions is the process of reducing them down to their simplest form. This means that both the numerator and the denominator have no common factors, other than 1.
For instance, after combining the numerators in our exercise, we arrived at the fraction \(-\frac{2}{4}\).

To simplify:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• In this case, \(-2\) and \(4\) have a GCD of \(2\).
• Divide both the numerator and the denominator by this GCD.

Simplifying \(-\frac{2}{4}\) by dividing both the top and bottom by \(2\) results in \(-\frac{1}{2}\).
Always check if a fraction can be simplified further. A fully simplified fraction is easier to understand and work with in subsequent problems.

Numerators and Denominators Explained

In any fraction, the numerator and denominator serve crucial roles. The numerator is the top number, showing how many parts you have or are dealing with.
The denominator, on the other hand, is the bottom number, indicating into how many pieces the whole has been divided.

In our task with \(-\frac{3}{4}\) \(+\) \(\frac{1}{4}\):

• The numerators \(-3\) and \(1\) tell us we're dealing with negative three parts and positive one part out of four equal parts.
• The denominator \(4\) is constant across both fractions, highlighting that they're in parts of identical size.

The sum is simplified by focusing on adding or subtracting the numerators, given the shared denominator, thus simplifying calculations.
Understanding numerators and denominators is key in fraction operations, as they define and limit how we work with fractional values.