Problem 70 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{4}{6}$$

Step-by-Step Solution

Answer

The reduced fraction is \( \frac{2}{3} \).

Step 1: Identify the Greatest Common Divisor

The first step is to determine the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \( \frac{4}{6} \), the numerator is 4 and the denominator is 6. Both 4 and 6 are divisible by 2. Thus, the GCD of 4 and 6 is 2.

Step 2: Divide the Numerator and Denominator by the GCD

Next, divide both the numerator and the denominator by their greatest common divisor. For \( \frac{4}{6} \), divide 4 and 6 by 2. This gives:\[ \frac{4 \div 2}{6 \div 2} = \frac{2}{3} \]

Step 3: Verify the Fraction is in Lowest Terms

Finally, check that the fraction is in its simplest form. The fraction \( \frac{2}{3} \) cannot be reduced further because 2 and 3 only share 1 as a common divisor. Thus, \( \frac{2}{3} \) is the fraction in its lowest terms.

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor, or GCD, is a crucial concept when it comes to reducing fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is the starting point for simplifying fractions. Let's take the fraction \( \frac{4}{6} \). Here, the numbers 4 and 6 need a common divisor. By listing their divisors:

• The divisors of 4 are 1, 2, and 4.
• The divisors of 6 are 1, 2, 3, and 6.

The largest number common to both lists is 2, making it the GCD.

Why is finding the GCD important? It determines how we scale down a fraction efficiently, ensuring fair comparisons and easy calculations. This process helps convert complex fractions into simpler forms.

Lowest Terms

Reducing a fraction to its lowest terms means rewriting it using the smallest possible numerator and denominator. After finding the GCD, you divide both the top and bottom numbers of the fraction by this number.

In our example \( \frac{4}{6} \), once the GCD (2) is identified:

• 4 divided by 2 gives 2.
• 6 divided by 2 gives 3.

This leads us to \( \frac{2}{3} \), which is the fraction in its simplest form.

Being in lowest terms makes fractions more interpretable and easier to work with, ensuring consistency across mathematical operations. It simplifies arithmetic calculations and allows for better comparison with other fractions.

Numerator and Denominator

Every fraction is made up of two parts: the numerator and the denominator. Understanding their roles is essential in fraction reduction.

- **Numerator**: The number on top of the fraction. It represents how many parts of a whole are being considered. For \( \frac{4}{6} \), the numerator is 4.- **Denominator**: The number on the bottom. It indicates the total number of equal parts the whole is divided into. For \( \frac{4}{6} \), the denominator is 6.

In the process of reducing fractions, these two components are analyzed for their common factors. We derived that both 4 and 6 can be divided by 2 without leaving a remainder, using the concept of the GCD as a guide.

Thus, understanding these elements helps one grasp why simplifying fractions alters them numerically but keeps the value equivalent, offering a clearer, more precise numerical expression.