Problem 12 - Fundamentals of Mathematics

Question

Reduce each fraction to lowest terms. \(\frac{12}{16}\)

Step-by-Step Solution

Answer

\(\frac{3}{4}\)

Step 1: Find the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, start by identifying the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \(\frac{12}{16}\), list the factors: - Factors of 12: 1, 2, 3, 4, 6, 12- Factors of 16: 1, 2, 4, 8, 16The greatest common factor in both lists is 4.

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

Next, divide both the numerator and the denominator of the fraction by the GCD found in Step 1. Divide 12 by 4 and 16 by 4:\[\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4}\]

Step 3: Verify the Simplified Fraction

Finally, confirm that the fraction \(\frac{3}{4}\) is in its simplest form. To verify, ensure that there are no common factors other than 1 between the numerator and the denominator. Since 3 and 4 have no other common factors, \(\frac{3}{4}\) is indeed simplified.

Key Concepts

Greatest Common Divisor

Understanding the greatest common divisor (GCD) is helpful when working with fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. To find the GCD:

• List all factors of the numerator.
• List all factors of the denominator.
• Identify the largest factor that appears in both lists.

For example, consider the fraction \(\frac{12}{16}\). The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 16 are 1, 2, 4, 8, and 16. The largest number common to both lists is 4, making it the GCD. Using the GCD simplifies the fraction to its smallest possible form, which leads to simplification.

Simplification

Simplification is the process of reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. This makes it easier to compare and understand fractions.

• First, find the GCD of the numerator and the denominator.
• Then, divide both the numerator and the denominator by this GCD.

In the example \(\frac{12}{16}\), after finding the GCD as 4, you divide both 12 and 16 by 4. This results in \(\frac{3}{4}\). Verifying the simplified result ensures no further simplification is possible, as 3 and 4 do not share any other factors than 1.

Numerator and Denominator

The numerator and denominator are key components of a fraction. The numerator is the top number, while the denominator is the bottom number. They help in dividing the whole into parts.

• The numerator indicates how many parts are considered.
• The denominator shows into how many equal parts the whole is divided.

In \(\frac{12}{16}\), 12 is the numerator, and 16 is the denominator. Each plays a crucial role in expressing the proportion of parts taken from a whole. To simplify, both numbers are divided by the GCD, leading to a clearer representation of the fraction.

Factors

Factors are numbers that multiply together to get another number. When finding factors:

• Consider all numbers that can divide another number without leaving a remainder.
• Every number has factors including 1 and itself.

For example, with 12, the factors are 1, 2, 3, 4, 6, and 12. Similarly, for 16, the factors are 1, 2, 4, 8, and 16. Knowing the factors of both the numerator and the denominator helps in identifying the GCD, which is key to simplifying fractions. By using factor lists, students can see commonalities and understand the basic building blocks of numbers.