Problem 31 - Basic College Mathematics

Question

Simplify. $$ \frac{17}{51} $$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{1}{3}\).

Step 1: Find the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{17}{51}\), first find the greatest common divisor (GCD) of the numerator (17) and the denominator (51). The GCD of 17 and 51 is 17 because 17 is a prime number and it divides both 17 and 51.

Step 2: Divide Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their GCD. The fraction becomes: \[ \frac{17 \div 17}{51 \div 17} = \frac{1}{3} \]

Step 3: Write the Simplified Fraction

After dividing both the numerator and the denominator by the GCD, the simplified fraction is \(\frac{1}{3}\).

Key Concepts

Greatest Common Divisor

In the process of simplifying fractions, finding the Greatest Common Divisor (GCD) is a crucial step. The GCD of two numbers is the largest number that evenly divides both of them. For example, in the fraction \(\frac{17}{51}\), we need the GCD of 17 and 51 to simplify the fraction correctly.

The GCD helps us reduce the numerator and denominator by the same amount, ensuring the fraction value remains the same. For our example, 17 is the GCD of both 17 and 51 since 17 is the highest number that can divide them evenly. This is because 17 is a prime number, which means its only divisors are 1 and itself. By dividing both the numerator and the denominator by 17, we get: \[ \frac{17 \div 17}{51 \div 17} = \frac{1}{3} \]

Prime Numbers

Understanding prime numbers is essential for finding the GCD. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They play a unique role in fraction simplification because they only have two factors.
Prime numbers between 1 and 100 include:

• 2
• 3
• 5
• 7
• 11
• 13
• 17
• 19
• 23
• and so on...

Using prime numbers can often make the process of identifying the GCD simpler. In the exercise given, 17 is a prime number, making it straightforward to recognize that it is the GCD of both 17 and 51. Thus, knowing prime numbers can give you an edge in math problems involving fractions.

Numerator and Denominator

Understanding the roles of the numerator and denominator in fractions helps you simplify them accurately. The numerator is the top number in a fraction, representing how many parts of a whole are being considered. In the fraction \(\frac{17}{51}\), 17 is the numerator.

The denominator, on the other hand, is the bottom number that tells you into how many parts the whole is divided. Here, 51 is the denominator.

• Numerator: The number above the fraction line
• Denominator: The number below the fraction line

To simplify a fraction, you need to divide both the numerator and the denominator by their GCD. When both are divided by the GCD, the fraction becomes smaller but retains the same value. After finding the GCD (which is 17 in our example), we divide both parts of our fraction: \[ \frac{17 \div 17}{51 \div 17} = \frac{1}{3} \]. The fraction \(\frac{1}{3}\) is the simplest form of \(\frac{17}{51}\).