Problem 73 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{14}{6}$$

Step-by-Step Solution

Answer

\( \frac{14}{6} \) reduces to \( \frac{7}{3} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, first find the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \( \frac{14}{6} \), the GCD of 14 and 6 is 2 because 2 is the largest integer that divides both numbers evenly.

Step 2: Divide Both the Numerator and Denominator by the GCD

Divide the numerator and the denominator by the GCD found in Step 1. For \( \frac{14}{6} \), divide both 14 and 6 by 2:\[\frac{14 \div 2}{6 \div 2} = \frac{7}{3}\]

Step 3: Verify the Fraction is in Lowest Terms

After dividing, check if the resulting fraction \( \frac{7}{3} \) can be simplified further. Since there are no common factors between 7 and 3 other than 1, the fraction is already in its simplest form.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

Knowing how to find the Greatest Common Divisor (GCD) is essential in simplifying fractions successfully. The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. Finding the GCD is a critical step when you want to simplify fractions.

• For example, consider the numbers 14 and 6.
• To find the GCD, list the divisors of each number.
• The divisors of 14 are: 1, 2, 7, and 14.
• The divisors of 6 are: 1, 2, 3, and 6.
• The greatest number that appears in both lists is 2.

Thus, the GCD of 14 and 6 is 2. Knowing how to find the GCD helps in making sure that when you simplify a fraction, it remains equivalent to the original fraction.

The Basics of Reducing Fractions

Reducing fractions is all about making them simpler while maintaining the same value. It's a bit like cleaning up a messy room! You essentially want to find a simpler version of the fraction by dividing both the numerator and denominator by their GCD.

• Using the example of \( \frac{14}{6} \), once we find the GCD is 2, we divide both 14 and 6 by this number.
• This gives us \( \frac{14 \div 2}{6 \div 2} = \frac{7}{3} \).
• Reducing fractions helps make calculations easier and more intuitive.

So, even though \( \frac{14}{6} \) and \( \frac{7}{3} \) represent the same value, the latter is more compact and easier to work with.

Recognizing the Simplest Form

Once you've divided a fraction as much as possible, you reach what's called the "simplest form". The simplest form of a fraction occurs when the numerator and the denominator have no common factors other than 1. This means that the fraction cannot be reduced any further.

• In the previous step, we reduced \( \frac{14}{6} \) to \( \frac{7}{3} \).
• We verify the simplest form by checking the GCD of 7 and 3.
• Since their only common factor is 1, \( \frac{7}{3} \) is indeed in its simplest form.

Having your fractions in simplest form ensures that answers are clear and concise, making mathematical operations like addition, subtraction, and comparison much simpler.