Problem 44 - Prealgebra

Question

Express each ratio or rate as a fraction in simplest form. 321 articles in 107 magazines

Step-by-Step Solution

Answer

The ratio is \( \frac{321}{107} \) in simplest form.

Step 1: Understand the Problem

We have 321 articles that are distributed across 107 magazines. We are asked to express this relationship in the form of a fraction, representing the ratio of articles to magazines.

Step 2: Write the Ratio as a Fraction

To express the ratio of 321 articles to 107 magazines, we write it as a fraction: \( \frac{321}{107} \).

Step 3: Simplify the Fraction

We need to check if the fraction \( \frac{321}{107} \) is in its simplest form. To do this, we must find the greatest common divisor (GCD) of 321 and 107. If the GCD is 1, then the fraction is already in simplest form.

Step 4: Calculate the GCD

Check the divisibility of 321 and 107. 107 is a prime number and doesn't divide 321 without a remainder. Therefore, the GCD is 1.

Step 5: Confirm the Simplest Form

Since the GCD of 321 and 107 is 1, the fraction \( \frac{321}{107} \) cannot be simplified further and is already in simplest form.

Key Concepts

Fraction Simplification

Simplifying a fraction means reducing it to its lowest terms, where the numerator (top number) and the denominator (bottom number) have no common factors other than 1. This makes the fraction as straightforward as possible and easier to understand.

• To simplify a fraction, you first need to identify any common factors shared by the numerator and the denominator.
• If such a factor exists, divide both the numerator and the denominator by this common factor.

For example, in our exercise, the fraction is \( \frac{321}{107} \). Our goal is to see if this fraction can be simplified by identifying any common factors. In this particular case, since the greatest common factor is 1, the fraction is already in its simplest form. Simplifying fractions not only helps in better understanding but also in performing arithmetic operations effortlessly.

Greatest Common Divisor

The Greatest Common Divisor (GCD), sometimes called the Greatest Common Factor (GCF), is the largest positive integer that divides two or more numbers without leaving a remainder. Finding the GCD is an essential step in simplifying fractions.

• One way to find the GCD is by listing the factors of each number and identifying the largest one they have in common.
• Another efficient method is using the Euclidean algorithm, which involves dividing and finding remainders.

In our exercise, we look for the GCD of 321 and 107. Since 107 is a prime number, it doesn't divide 321 evenly. Therefore, their only common divisor is 1, establishing that \( \frac{321}{107} \) is already as simple as it can get.Understanding GCD is important because it helps ensure fractions are expressed in their simplest, most efficient form.

Prime Numbers

Prime numbers are special numbers greater than 1 that have no divisors other than 1 and themselves. They play a crucial role in number theory and in simplifying fractions.

• For a number to be prime, it must be divisible only by 1 and itself, making it impossible to be divided evenly by any other numbers.
• Prime numbers help in determining the GCD, as a prime divisor simplifies the process of checking divisibility.

In the context of our exercise, 107 is a prime number. Its primality simplifies the process of confirming the GCD with 321, as primes inherently factorize different sets of numbers, frequently simplifying findings like these. Recognizing prime numbers is useful in a variety of mathematical operations, including simplifying fractions, as they limit potential common divisors.