Problem 16 - Elementary Algebra

Question

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{32}{40}\)

Step-by-Step Solution

Answer

Question: Reduce the fraction 32/40 to its lowest terms. Answer: The reduced fraction is 4/5.

Step 1: Find the Greatest Common Divisor (GCD)

We need to find the greatest common divisor of the numerator (32) and the denominator (40). We can use the Euclidean Algorithm to compute the GCD: 1. Divide 40 by 32: The quotient is 1 and the remainder is 8. 2. Now, divide 32 by the remainder obtained in the previous step (8): The quotient is 4 and the remainder is 0. 3. Since the remainder is 0, the GCD of 32 and 40 is the divisor in the last step, which is 8.

Step 2: Reduce the fraction to its lowest terms

Now that we have the GCD of the numerator and the denominator (8), it's time to reduce the given fraction to its lowest terms. To do this, we simply divide the numerator and the denominator by the GCD: - Divide 32 by 8: The result is 4. - Divide 40 by 8: The result is 5. Hence, the given fraction \(\frac{32}{40}\) reduced to its lowest terms is \(\frac{4}{5}\).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that can divide two or more integers without leaving a remainder. In this exercise, the GCD is key to reducing fractions to their simplest form.
To find the GCD of two numbers, we check the common divisors each number has and identify the largest one. This helps streamline operations like simplifying fractions since it's the biggest factor shared between the numerator and the denominator.

• Finding the GCD efficiently simplifies calculations.
• It avoids long division and reduces fractions quickly.

The GCD can be found using various methods, with the Euclidean algorithm being one of the most popular and effective ways.

Euclidean Algorithm

The Euclidean Algorithm is a method for efficiently finding the greatest common divisor (GCD) of two numbers. It's based on the principle that the GCD of two numbers is the same as the GCD of their remainder when divided. This algorithm is fast and uses simple subtraction or modulo operations to break down the numbers into smaller pairs.
Here's how it works step-by-step:

• First, divide the larger number by the smaller one.
• Take note of the remainder. If the remainder isn't zero, repeat the process, using the smaller number as the new divisor and the remainder as the new dividend.
• Continue until the remainder is zero. At this point, the divisor from the last operation will be the GCD.

For example, in finding the GCD of 32 and 40, we divided 40 by 32, yielding a remainder of 8. We then divided 32 by 8 and achieved a remainder of 0, determining 8 as the greatest common divisor.

Lowest Terms

Reducing a fraction to its lowest terms involves dividing both the numerator and the denominator by their greatest common divisor (GCD). The idea is to express the fraction in its simplest form, where no number other than 1 can evenly divide both the top and bottom of the fraction.
This operation is crucial for many mathematical calculations and often required in tests and homework to show a clean, simplified result. To reduce a fraction:

• Start with the GCD obtained from the Euclidean Algorithm.
• Divide both the numerator and the denominator by this GCD.
• The outcome will be a fraction in its lowest terms.

In our example, after finding the GCD of 32 and 40 is 8, we divide these numbers by 8 to get the simplified fraction \(\frac{4}{5}\). This fraction cannot be simplified further, ensuring it's in the simplest form.