Problem 10 - Elementary Algebra

Question

For exercises 1-12, simplify. $$ \frac{5}{415} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{1}{83} \).

Step 1: Identify Greatest Common Divisor (GCD)

First, find the greatest common divisor (GCD) of the numerator (5) and the denominator (415). The GCD is the largest number that can divide both 5 and 415 without leaving a remainder. Since 5 is a prime number, we check if 5 is a divisor of 415.

Step 2: Divide by GCD

Since 5 is indeed a divisor of 415 (because 415 divided by 5 equals 83), divide both the numerator and the denominator by 5. \[ \frac{5 \div 5}{415 \div 5} = \frac{1}{83} \]

Step 3: Simplify the Fraction

The simplified fraction is obtained by completing the division: \[ \frac{5}{415} = \frac{1}{83} \]

Key Concepts

greatest common divisor

When simplifying fractions, one method involves finding the greatest common divisor (GCD). The GCD is the highest number that divides two or more numbers without leaving any remainder. It helps reduce the fraction to its simplest form. For example, in the fraction \(\frac{5}{415}\), we need to find the GCD of 5 and 415. Since 5 is a prime number, it is only divisible by 1 and itself. To check if 5 is a divisor of 415, we perform the division: 415 divided by 5 equals 83 with no remainder. Thus, 5 is the GCD. By dividing both the numerator (5) and the denominator (415) by the GCD (5), we simplify the fraction to \(\frac{1}{83}\).

prime numbers

Prime numbers play a critical role in simplifying fractions. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. These include numbers like 2, 3, 5, 7, and so on. In the exercise, 5 is identified as a prime number. This uniqueness means it can only be divided by 1 and 5. When the numerator of a fraction is a prime number, simplifying the fraction often becomes more straightforward, especially if that prime number turns out to be the GCD with the denominator, as in the fraction \(\frac{5}{415}\). By recognizing and using prime numbers, we can efficiently reduce fractions and grasp the underlying concept of simplicity in fractions.

numerator and denominator

In any fraction, there are two main components: the numerator and the denominator. The numerator is the top number, which represents the part of the whole. The denominator is the bottom number, indicating how many equal parts the whole is divided into. For example, in the fraction \(\frac{5}{415}\), 5 is the numerator and 415 is the denominator. Understanding how these parts interact is crucial for operations like simplifying fractions. The goal in simplifying is to find an equivalent fraction where the numerator and denominator are as small as possible. This process often involves finding the greatest common divisor (GCD) and using it to divide both the numerator and the denominator, making the fraction easier to understand and work with. For our example, by dividing both 5 and 415 by their GCD (5), we get \(\frac{1}{83}\), which represents the simplified form of the original fraction.