Problem 13 - Fundamentals of Mathematics

Question

Reduce each fraction to lowest terms. \(\frac{9}{24}\)

Step-by-Step Solution

Answer

\( \frac{3}{8} \)

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{9}{24} \) to its lowest terms, we first need to determine the greatest common divisor (GCD) of the numerator (9) and the denominator (24).

Step 2: Find the Factors

List all the factors of 9 and 24. The factors of 9 are \(1, 3, 9\) and the factors of 24 are \(1, 2, 3, 4, 6, 8, 12, 24\).

Step 3: Determine the GCD

Identify the largest common factor between 9 and 24. From the lists, the common factors are \(1, 3\) and the greatest of these is \(3\).

Step 4: Divide Numerator and Denominator by GCD

Divide both the numerator and the denominator by the GCD. Divide 9 by 3 to get 3 and divide 24 by 3 to get 8. Thus, \( \frac{9}{24} \) reduces to \( \frac{3}{8} \).

Step 5: Confirm the Reduction

Verify that \( \frac{3}{8} \) is in its simplest form by checking that 3 and 8 have no common factors other than 1.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is one of the fundamental concepts in reducing fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction.
To find the GCD, follow these simple steps:

• Write down all factors of the numerator (the top number of your fraction).
• Do the same for the denominator (the bottom number).
• Identify the common factors shared by both numbers.
• Choose the largest factor common to both lists. This is your GCD.

For example, when reducing the fraction \( \frac{9}{24} \), we found that the factors of 9 are \(1, 3, 9\), and for 24, the factors are \(1, 2, 3, 4, 6, 8, 12, 24\). Both numbers share the factors \(1\) and \(3\), with \(3\) being the greatest. Using the GCD ensures that we can simplify our fraction efficiently.

Achieving the Lowest Terms

Reducing a fraction to its lowest terms means simplifying it so that the numerator and denominator are the smallest possible whole numbers. This is accomplished by dividing both by their greatest common divisor.
Here's how to check if a fraction is in its lowest terms:

• Ensure the only common factor between the numerator and the denominator is 1.
• Reassess the fraction after simplifying to double-check no further simplification is possible.

Taking \( \frac{9}{24} \) as an example, after identifying the GCD as 3, we divide both 9 and 24 by this number. This gives us \( \frac{3}{8} \), which cannot be simplified further because 3 and 8 have no common factors apart from 1. Therefore, \( \frac{3}{8} \) is in its lowest terms.

Importance of Mathematics Education in Fraction Reduction

Understanding how to reduce fractions is a key skill in mathematics education. It forms the backbone for more advanced concepts such as solving equations and working with ratios. Students often begin learning about fractions early on, and reducing fractions teaches them to:

• Organize and compare numbers effectively.
• Develop logical thinking skills.
• Apply mathematical concepts to real-life scenarios.

Mathematics education empowers students not just to handle numbers, but to see the patterns within them. This foundational understanding assists students beyond math class, fostering analytical skills they will use throughout their education and in daily life. When learners master concepts like the greatest common divisor, they improve their problem-solving capabilities, equipping them for tasks that require critical thinking and precise calculations.