Problem 65 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{4}{14}$$

Step-by-Step Solution

Answer

The fraction \( \frac{4}{14} \) reduced to its lowest terms is \( \frac{2}{7} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, determine the greatest common divisor of the numerator and the denominator. For this exercise, we need to find the GCD of 4 and 14.

Step 2: Calculate the GCD

The factors of 4 are 1, 2, and 4. The factors of 14 are 1, 2, 7, and 14. The common factors between 4 and 14 are 1 and 2. Therefore, the GCD is 2.

Step 3: Simplify the Fraction

Divide both the numerator and the denominator of \( \frac{4}{14} \) by their GCD, which is 2. This gives \( \frac{4 \div 2}{14 \div 2} = \frac{2}{7} \).

Key Concepts

Greatest Common Divisor

When reducing fractions, one of the first steps is finding the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest positive integer that divides both numbers without leaving a remainder. This is important because it helps in simplifying the fraction to its lowest terms.

To determine the GCD, list out all factors of both the numerator and the denominator. For example, with the fraction \( \frac{4}{14} \), the factors of 4 are 1, 2, and 4, while the factors of 14 are 1, 2, 7, and 14. The common factors are 1 and 2. The greatest of these is 2, so the GCD is 2.

• Steps to find the GCD:
  • List factors of each number.
  • Identify common factors.
  • Select the largest common factor.

Finding the GCD is crucial for simplifying fractions effectively. It makes the rest of the process straightforward.

Numerator and Denominator

In a fraction, the numerator is the top number, and the denominator is the bottom number. Understanding the roles of these components is essential when working with fractions.

• The numerator indicates how many parts of the whole you have. In our example \( \frac{4}{14} \), the numerator is 4.
• The denominator tells you into how many total parts the whole is divided. In \( \frac{4}{14} \), the denominator is 14.

Knowing how to manage these terms effectively is vital, especially when reducing fractions. When reducing, we maintain the same proportion by dividing both the numerator and the denominator by the same number, often the GCD. This operation changes their individual values but leaves the overall value of the fraction unchanged. It's very much like slicing a pizza; each piece doesn't get smaller, you just have fewer of them to manage together.

Simplifying Fractions

Once you have determined the GCD of the numerator and the denominator, simplifying the fraction becomes a straightforward task. Simplifying a fraction means rewriting it in simpler terms but with the same value.

To simplify \( \frac{4}{14} \), divide both 4 (numerator) and 14 (denominator) by their GCD, which is 2. Doing this, you get:\[\frac{4 \div 2}{14 \div 2} = \frac{2}{7}\]This fraction, \( \frac{2}{7} \), is now in its simplest form, as there is no common divisor bigger than 1 left to further reduce it.

• Steps for Simplifying:
  • Calculate the GCD of numerator and denominator.
  • Divide both the numerator and the denominator by the GCD.
  • Write down the new fraction.

Simplifying fractions ensures that they are easier to work with in mathematical operations, presenting the smallest possible whole numbers that maintain the ratio represented by the original fraction.