Problem 8 - Fundamentals of Mathematics

Question

Reduce each fraction to lowest terms. $$\frac{6}{48}$$

Step-by-Step Solution

Answer

The fraction reduces to \(\frac{1}{8}\).

Step 1: Find the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{6}{48} \), we need to find the greatest common divisor (GCD) of the numerator (6) and the denominator (48). List the factors of both numbers. Factors of 6: 1, 2, 3, 6 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 The common factors are 1, 2, 3, and 6. The greatest of these is 6.

Step 2: Divide Numerator and Denominator by the GCD

Now that we've found the GCD is 6, divide both the numerator and the denominator of the fraction by this number. \[ \frac{6 \div 6}{48 \div 6} = \frac{1}{8} \]

Step 3: Check the Reduced Fraction

Ensure that the fraction \(\frac{1}{8}\) cannot be simplified further. Look for common factors between the numerator and the denominator. Factors of 1: 1 Factors of 8: 1, 2, 4, 8 The only common factor is 1, so \(\frac{1}{8}\) is already in its simplest form.

Key Concepts

Greatest Common Divisor

When simplifying fractions, one important concept is the Greatest Common Divisor (GCD). This is the largest number that divides both the numerator and the denominator without leaving a remainder. It helps in reducing a fraction to its simplest form.
Here's how to find the GCD:

• List all factors of the numerator.
• List all factors of the denominator.
• Identify the largest number common to both lists.

For example, in the fraction \( \frac{6}{48} \), the factors of 6 are 1, 2, 3, and 6. The factors of 48 include 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The greatest number in both lists is 6, making it the GCD. Finding this number is crucial for breaking down fractions efficiently.

Numerator and Denominator

A fraction consists of two main parts: the numerator and the denominator. The numerator is the top number and represents how many parts of the whole we are considering. In contrast, the denominator is the bottom number and it shows into how many equal parts the whole is divided.
Let's look at \( \frac{6}{48} \):

• The numerator is 6, which means there are 6 parts considered.
• The denominator is 48, indicating those parts are out of a total of 48 equal parts.

By understanding these components, you can easily identify what part of the fraction needs attention when simplifying. The goal is to make these parts represent the same ratio but with smaller numbers.

Simplest Form

The simplest form of a fraction occurs when the greatest common divisor (GCD) between its numerator and denominator is 1. This means the numbers in the fraction have been divided by their GCD, leaving a simplified and more manageable form.
In our example, \( \frac{6}{48} \) was simplified:

• We found the GCD, which was 6.
• Divided both numerator and denominator by this GCD: \( 6 \div 6 = 1 \) and \( 48 \div 6 = 8 \).

So, the simplified (or simplest) form of \( \frac{6}{48} \) is \( \frac{1}{8} \). Always check that there are no common factors other than 1, ensuring the fraction is indeed in its simplest form. This approach simplifies calculations and makes understanding fractions easier.