Problem 14 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$ 20003000 $$

Step-by-Step Solution

Answer

The fraction \( \frac{2000}{3000} \) reduces to \( \frac{2}{3} \).

Step 1: Analyze the Fraction

The given fraction is \( \frac{2000}{3000} \). To reduce it to the lowest terms, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

Step 2: Find the GCD

First, let's list the prime factors of 2000 and 3000. - 2000 can be factored into: \( 2^4 \times 5^3 \).- 3000 can be factored into: \( 2^3 \times 3 \times 5^3 \).The common factors are \( 2^3 \) and \( 5^3 \). Thus, the GCD is \( 2^3 \times 5^3 = 8 \times 125 = 1000 \).

Step 3: Divide by the GCD

Once we have the GCD, we divide both the numerator and the denominator by 1000. \[ \frac{2000}{3000} = \frac{2000 \div 1000}{3000 \div 1000} = \frac{2}{3} \]

Step 4: Verify the Result

Make sure \( \frac{2}{3} \) is in its lowest terms. Since 2 and 3 are both prime numbers, the fraction is simplified completely.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is an essential tool in reducing fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction.
By finding the GCD, you can simplify the fraction to its lowest terms, making it easier to understand and work with.
To find the GCD:

• List the factors for both numerator and denominator.
• Identify the common factors.
• Choose the greatest one.

For example, with the fraction \( \frac{2000}{3000} \), the gcd is 1000, as it is the largest number that divides both 2000 and 3000 without leaving a remainder.
This method ensures that a fraction is expressed in its simplest form, which is crucial for making efficient calculations and comparisons.

Prime Factorization of Numbers

Prime factorization involves breaking down a number into its smallest prime components. This method is very effective when searching for the GCD. It helps to accurately identify the common factors between two numbers.
Here's how you prime factorize a number:

• Start by dividing the number by the smallest prime number (2).
• Continue dividing by the same prime until it no longer divides evenly.
• Move to the next smallest prime number (3, 5, etc.).
• Repeat the process until the remainder is a prime number itself.

In the case of the numbers 2000 and 3000:- The prime factorization of 2000 is \( 2^4 \times 5^3 \).- The prime factorization of 3000 is \( 2^3 \times 3 \times 5^3 \).
This breakdown helps in identifying the common prime factors \( 2^3 \) and \( 5^3 \), leading to the GCD of 1000. Understanding this method aids in optimizing fraction reduction.

Reducing Fractions to Lowest Terms

Reducing a fraction to its lowest terms means simplifying it as much as possible so that the numerator and the denominator have no common factors other than 1.
Here’s the step-by-step process:

• Determine the GCD of the numerator and denominator.
• Divide both the numerator and denominator by this GCD.

For instance, given \(\frac{2000}{3000}\), after finding the GCD to be 1000, you would divide both 2000 and 3000 by 1000.
This results in \(\frac{2}{3}\), which is the fraction in its lowest terms.
It is important to check at the end that the new numerator and denominator are coprime—that is, they share no further common divisors.
This ensures the fraction is truly in its simplest form, facilitating easier computation and understanding in mathematics.