Problem 9 - Prealgebra

Question

Find the following sums and differences, and reduce to lowest terms. (Add or subtract as indicated.) $$\frac{1}{4}+\frac{2}{4}+\frac{3}{4}$$

Step-by-Step Solution

Answer

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

Step 1: Add the Numerators

The denominators of all the fractions are the same (4), so we can add the numerators directly. The numerators are 1, 2, and 3, so we calculate: \[1 + 2 + 3 = 6\]

Step 2: Form the Fraction

Since the sum of the numerators is 6 and the common denominator is 4, we form the fraction: \[\frac{6}{4}\]

Step 3: Reduce the Fraction

To reduce the fraction \(\frac{6}{4}\) to its lowest terms, find the greatest common divisor (GCD) of 6 and 4, which is 2. Divide both the numerator and the denominator by their GCD:\[\frac{6\div2}{4\div2} = \frac{3}{2}\]

Key Concepts

Numerator Addition

Adding fractions is simple when they have the same denominator. We can focus directly on the numerators when this condition is met. When you see fractions like \( \frac{1}{4}, \frac{2}{4}, \) and \( \frac{3}{4}, \) pay attention to the top numbers. The common denominator, which we'll discuss next, allows us to directly add these numerators.
To add fractions with the same denominator, just add the numerators:

• Take the numerator from each fraction.
• Add them together.
• Keep the same denominator.

In our example, you have numerators 1, 2, and 3:

• Numerator addition looks like this: \[1 + 2 + 3 = 6\]
• Place this sum over the common denominator.

This approach leads to forming the fraction \(\frac{6}{4}.\) It’s important to keep track of both the numerators and whether or not the denominators match to make the addition process seamless.

Common Denominator

The common denominator is a crucial aspect when adding fractions. It lets you easily combine fractions since you're effectively dealing with "like" units. In our problem, the denominator is 4 in each fraction, thus all fractions share the common denominator. This consistency is what makes adding the numerators straightforward.
If fractions have different denominators, you'd need to make an adjustment to find a common one before adding or subtracting:

• Identify the least common multiple of all denominators.
• Convert each fraction so that they all share this common denominator.
• Continue with the addition as you did with like denominators.

In our case, because \(\frac{1}{4}, \frac{2}{4}, \) and \(\frac{3}{4}\) share the same denominator already, it simplifies the entire process. The common denominator often aids in reducing complexity when dealing with fractional operations, helping ensure sums and differences are accurate.

Reducing Fractions

After adding fractions, it's vital to simplify the resulting fraction. This process, known as reducing, aims to express the fraction in its simplest form. To achieve this, find the greatest common divisor (GCD) of the numerator and the denominator.
Here's a quick method:

• Identify the GCD between the numerator and the denominator.
• Divide both the numerator and denominator by this GCD.

For the fraction \(\frac{6}{4},\) the GCD of 6 and 4 is 2:

• Divide the numerator and denominator by 2: \[\frac{6 \div 2}{4 \div 2} = \frac{3}{2}\]
• Check that the numerator and denominator no longer share any common factors other than 1.

Once reduced, \(\frac{3}{2}\) represents the simplest form of the fraction. Simply put, reducing fractions makes them easier to interpret and ensures you're expressing values concisely.