Problem 1 - Prealgebra

Question

Find the following sums and differences, and reduce to lowest terms. (Add or subtract as indicated.) $$\frac{3}{6}+\frac{1}{6}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Common Denominator

The given fractions are \(\frac{3}{6}\) and \(\frac{1}{6}\). Both fractions have the same denominator, which is 6. Therefore, we can directly add the numerators.

Step 2: Add the Numerators

Since the denominator is the same, add the numerators: \(3 + 1 = 4\). So, the sum is \(\frac{4}{6}\).

Step 3: Simplify the Fraction

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

Key Concepts

Common Denominator

When adding or subtracting fractions, the fractions need a common denominator. This is because fractions represent parts of a whole, and to accurately add them together, they must be out of the same total (denominator).
A common denominator is simply a shared denominator between fractions. It's essentially the number that both fractions’ denominators can divide into without a remainder.
For instance, in the exercise \[\frac{3}{6} + \frac{1}{6}\], both fractions already share the denominator 6. This makes it easy to proceed directly to adding their numerators.

• Identify or determine the shared denominator.
• Adjust only the numerators when denominators are the same.
• If the denominators aren't the same, find a least common multiple (LCM) of the denominators to use as the new denominator.

Remember: A common denominator simplifies the process of adding and subtracting fractions. Align the fractions to the same base, and you can solve them much more straightforwardly.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their lowest terms. This means expressing the fraction in such a way that the numerator and the denominator have no common divisors other than 1. This simplified form is always preferred for its clarity and elegance.
To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that evenly divides both the numerator and the denominator.Consider the fraction from our example \(\frac{4}{6}\):

• The GCD of 4 and 6 is 2.
• Divide both numerator and denominator by 2: \[\frac{4 \div 2}{6 \div 2} = \frac{2}{3}\]

Now, the fraction \(\frac{2}{3}\) is simplified because there is no number other than 1 that divides both 2 and 3. Simplified fractions are easier to understand and work with in further calculations.

Greatest Common Divisor

To simplify a fraction, finding the greatest common divisor (GCD) is essential. The GCD of two numbers is the greatest number that can divide both without leaving a remainder. It is a vital part of simplifying fractions.
Finding the GCD can be accomplished in various ways:

• **Prime Factorization:** Break down both numbers into their prime factors and multiply the shared factors. For example, 4 factors into 2 x 2 and 6 factors into 2 x 3. The shared factor is 2.
• **Euclidean Algorithm:** A more advanced, step-by-step subtraction-based approach for larger numbers.

In the exercise, the GCD of the numerator 4 and the denominator 6 was found to be 2, thus simplifying \(\frac{4}{6}\) to \(\frac{2}{3}\). Identifying the GCD simplifies fractions efficiently, ensuring calculations remain easy and clear.