Problem 61 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. (Skills Review p. 763) $$\frac{53}{424}$$

Step-by-Step Solution

Answer

The simplest form of \( \frac{53}{424} \) is \( \frac{53}{424} \).

Step 1: Find the GCD

Find the greatest common divisor (GCD) of 53 and 424. To do this, list out all the divisors of both numbers and find the largest one that divides both numbers. The GCD of 53 and 424 is 1.

Step 2: Divide the Numerator and the Denominator

To simplify the fraction, divide both the numerator and denominator by their GCD. However, since the GCD is 1, dividing by 1 does not change the numbers. Thus, \( \frac{53}{424} \) is the simplest form of the fraction.

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor (GCF), of two numbers is the largest number that divides both of them without leaving a remainder. It's a cornerstone concept for simplifying fractions, as it allows us to reduce them to their simplest form.

For instance, to find the GCD of two numbers, we usually list all the divisors of each number, then identify the largest one they have in common. In our exercise, the numbers in question are 53 and 424. Since 53 is a prime number, it only has two divisors: 1 and itself. On the other hand, 424 has several divisors, but the only one it shares with 53 is 1. Therefore, the GCD of 53 and 424 is 1.

Knowing how to calculate the GCD is essential, especially when dealing with larger numbers where it might not be as obvious. Sometimes, using the Euclidean algorithm can simplify this process significantly, especially for numbers that are not easily factorable.

Numerator and Denominator

A fraction is comprised of two parts: the numerator and the denominator. The numerator, located above the fraction bar, represents the number of parts we have. The denominator, below the fraction bar, indicates the total number of equal parts that make up a whole.

When simplifying fractions, our goal is to reduce the numerator and the denominator to their smallest possible values while keeping the same value of the fraction. Dividing both by their GCD is the standard method of doing this. For our example, \( \frac{53}{424} \), 53 is the numerator and 424 is the denominator.

The process for simplifying involves dividing each by their GCD, which, in this case, is 1. Dividing any number by 1 leaves the number unchanged, so the fraction remains the same. Understanding how numerators and denominators work together helps us interpret fractions and manipulate them algebraically.

Simplest Form of a Fraction

The simplest form of a fraction, also known as its reduced form, occurs when the numerator and the denominator are as small as possible and no further reduction is possible – meaning there is no common divisor between them other than 1. A fraction is in simplest form when the GCD of its numerator and denominator is 1.

The fraction \( \frac{53}{424} \) is already in the simplest form because the GCD of 53 and 424 is 1, indicating they have no other common divisors. This is an important concept in mathematics, as working with fractions in their simplest form often makes calculation and algebraic manipulations much easier.

Although our example was already in simplest form, many fractions are not and need to be simplified. To do so, divide both the numerator and the denominator by their GCD. If the GCD is already 1, the fraction cannot be simplified further. Always remember to check for the simplest form to ensure the clarity of your mathematical expressions.