Problem 11 - Elementary Algebra

Question

For exercises 1-12, simplify. $$ \frac{34}{51} $$

Step-by-Step Solution

Answer

\( \frac{2}{3} \)

Step 1 - Find the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{34}{51} \), find the greatest common divisor (GCD) of the numerator (34) and the denominator (51). The prime factors of 34 are 2 and 17, and the prime factors of 51 are 3 and 17. The common prime factor is 17.

Step 2 - Divide the Numerator by the GCD

Divide the numerator (34) by the GCD (17). Simplification: \[ \frac{34}{17} = 2 \]

Step 3 - Divide the Denominator by the GCD

Divide the denominator (51) by the GCD (17). Simplification: \[ \frac{51}{17} = 3 \]

Step 4 - Write the Simplified Fraction

After dividing both the numerator and the denominator by the GCD, the simplified fraction is: Simplified fraction: \[ \frac{2}{3} \]

Key Concepts

Greatest Common Divisor (GCD)

To simplify any fraction properly, finding the Greatest Common Divisor (GCD) is crucial. The GCD of two numbers is the largest number that can divide both of them without leaving a remainder. For the fraction \(\frac{34}{51}\), the GCD helps us reduce it as much as possible. In this case, the GCD is the common prime factor shared by both the numerator and the denominator. By identifying the GCD first, we set the stage for effective simplification.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. For instance, to simplify the fraction \(\frac{34}{51}\), we need to factor both 34 and 51 into primes:

- The prime factors of 34 are 2 and 17 (since 34 = 2 × 17).
- The prime factors of 51 are 3 and 17 (since 51 = 3 × 17).

By identifying the common prime factor, we can determine the GCD. Common prime factors in 34 and 51 are crucial as they dictate how much we can reduce the fraction. Here, both numbers share the prime factor 17, which becomes our key to simplifying the fraction effectively.

Numerator and Denominator

Understanding the roles of the numerator and denominator is essential for simplifying fractions. The numerator is the top part of the fraction, while the denominator is the bottom part.

For \(\frac{34}{51}\), the numerator is 34, and the denominator is 51. To simplify the fraction, we need to divide both parts by the GCD (17). This gives us:

- Simplified numerator: \(\frac{34}{17} = 2\).
- Simplified denominator: \(\frac{51}{17} = 3\).

After this division, we get the simplified fraction \(\frac{2}{3}\). Understanding how the numerator and denominator interact with the GCD ensures a precise and simplified result.