Problem 42 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. $$ \frac{48}{64} $$

Step-by-Step Solution

Answer

The simplest form of the fraction \(\frac{48}{64}\) is \(\frac{3}{4}\).

Step 1: Find the GCD of the numerator and the denominator

The GCD of 48 and 64 can be found using the Euclidean algorithm or by listing the factors of each and finding the largest common factor. In this case, the GCD is 16.

Step 2: Divide the numerator and the denominator by the GCD

Dividing both the numerator 48 and the denominator 64 by the GCD 16, we get \(\frac{48}{16} = 3\) and \(\frac{64}{16} = 4\). So, the simplified form of the given fraction is \(\frac{3}{4}\).

Key Concepts

Greatest Common Divisor (GCD)

The term Greatest Common Divisor (GCD) refers to the highest number that evenly divides two or more integers without leaving a remainder. It's a foundational concept in mathematics, particularly in number theory, where it plays a crucial role in simplifying fractions, factoring, and solving problems related to divisibility.

Identifying the GCD helps in reducing fractions to their simplest form – a form where both the numerator and the denominator are as small as possible, yet still have the same value as the original fraction. For instance, when simplifying the fraction \( \frac{48}{64} \), the GCD of 48 and 64 is 16. This shared divisor is what makes it possible to reduce the fraction down to a simpler equivalent, \( \frac{3}{4} \).

Finding the GCD can be done in various ways including prime factorization or, more efficiently, using the Euclidean algorithm. Regardless of the method, understanding and calculating the GCD is a crucial step in working with fractions and other aspects of arithmetic.

Euclidean Algorithm

The Euclidean algorithm is a method used to find the Greatest Common Divisor (GCD) of two numbers, which is especially useful for simplifying fractions. It's an ancient, efficient process that involves a series of divisions. At its core, the algorithm is based on the principle that the GCD of two numbers also divides their difference.

To use the Euclidean algorithm, one begins with the two numbers in question (say, a and b where a > b). The process involves dividing a by b, taking note of the remainder, and then repeating this process by dividing b by the new remainder. This continues until the remainder is zero, at which point, the divisor from the last division is the GCD.

For our fraction \( \frac{48}{64} \), we would start by dividing 64 by 48, producing a remainder of 16. We then divide 48 by 16, receiving a remainder of zero, indicating that 16 is the GCD. This is a systematic and reliable way to find the GCD of any two numbers, irrespective of their size, making it invaluable in simplifying fractions and other mathematical operations.

Simplify Fractions Step by Step

To simplify a fraction step by step, one must follow a clear set of procedures that result in a fraction represented in its most basic terms. Simplifying fractions is a vital skill, making them easier to understand and work with, especially when performing addition, subtraction, or comparison of fractions.

The first step is identifying the GCD of the numerator and the denominator. Once the GCD is found, both the numerator and the denominator are divided by this number. This process ensures that the simplified fraction is mathematically equivalent to the original fraction, just expressed in smaller, more manageable terms.

For example, given the fraction \( \frac{48}{64} \), after determining that the GCD is 16, we divide both 48 and 64 by 16, yielding \( \frac{3}{4} \). This simplified fraction is much more straightforward to work with and still represents the same quantity as the original fraction. By reducing fractions, mathematical expressions and equations become neater, often leading to easier calculation and better understanding of relationships between numbers.