Problem 30 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{12}{84}$$

Step-by-Step Solution

Answer

\(\frac{12}{84}\) reduces to \(\frac{1}{7}\).

Step 1: Identify the Greatest Common Factor (GCF)

To reduce a fraction to its lowest terms, one must first determine the greatest common factor (GCF) of the numerator and the denominator. For the given problem \(\frac{12}{84}\), list the factors of each. Factors of 12 are 1, 2, 3, 4, 6, and 12. Factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84. The greatest common factor here is 12.

Step 2: Divide Both Numerator and Denominator by the GCF

After finding the GCF, which is 12, divide both the numerator (12) and the denominator (84) by this number. So we have \(\frac{12}{12} = 1\) and \(\frac{84}{12} = 7\).

Step 3: Write the Reduced Fraction

The result of dividing both the numerator and denominator by their greatest common factor gives us the fraction in its lowest terms: \(\frac{1}{7}\). Hence, \(\frac{12}{84}\) in its simplest form is \(\frac{1}{7}\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a crucial concept in reducing fractions. It refers to the largest number that can exactly divide both the numerator and the denominator of a fraction without leaving a remainder. Identifying the GCF is the first step to simplifying fractions.
To find the GCF for any two numbers:

• List out all of the factors for each number.
• Find the largest factor that appears in both lists.

In our exercise, we had the fraction \( \frac{12}{84} \). For the number 12, the factors were 1, 2, 3, 4, 6, and 12. For 84, the factors spanned from 1 to 84, including numbers like 6, 7, 12, 21, and 42. By comparing these lists, we note that 12 is the highest number that appears in both, making it the greatest common factor.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, is the process of converting a fraction to its simplest form. This involves reducing both the numerator and the denominator by their greatest common factor. Simplified fractions are more straightforward and often easier to work with.
To simplify a fraction, follow these steps:

• Find the greatest common factor (GCF) of the numerator and denominator.
• Divide both the numerator and the denominator by this GCF.

In our example \( \frac{12}{84} \), the GCF was 12. Dividing the numerator (12) by 12 gives 1, and dividing the denominator (84) by 12 gives 7.
Thus, the fraction simplifies to \( \frac{1}{7} \), meaning this is the simplest form of the original fraction.

Numerator and Denominator

Every fraction consists of two key parts: the numerator and the denominator. Understanding these components is essential for anyone learning how to work with fractions.
The numerator is the top number in a fraction and indicates how many parts of a whole are being considered. In our original example \( \frac{12}{84} \), 12 is the numerator, which represents the number of parts being taken.
The denominator is the bottom number in a fraction and shows how many equal parts the whole is divided into. For \( \frac{12}{84} \), 84 is the denominator, reflecting the total number of equal parts making up a whole.
Knowing the roles of these components helps in simplifying fractions, as the process involves deciding how to decrease these numbers while maintaining the fraction's value relatively unchanged. This is achieved through division by the greatest common factor, resulting in a simplified yet equivalent fraction.