Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each fraction by reducing it to its lowest terms. $$\frac{38}{50}$$

Step-by-Step Solution

Answer

The simplified form of the fraction \( \frac{38}{50} \) is \( \frac{19}{25} \).

Step 1: Identify the Greatest Common Divisor

Find the greatest common divisor (GCD), also known as greatest common factor, of 38 and 50. The GCD is the largest number that can evenly divide both numbers. In this case, the GCD is 2.

Step 2: Divide the Numerator and Denominator by the GCD

Next, divide both the numerator (38) and the denominator (50) by the GCD (2). This will result in: \( \frac{38}{2} / \frac{50}{2} \) which simplifies to \( \frac{19}{25} \).

Step 3: Check the Simplified Fraction

Finally, check if the fraction \( \frac{19}{25} \) can be further reduced to lower terms. Since 19 and 25 have no common factors other than 1, this is the lowest term of the fraction.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in mathematics, especially when it comes to simplifying fractions. It refers to the largest integer that can evenly divide two numbers without leaving a remainder. For instance, in the fraction \( \frac{38}{50} \), we need to find the GCD of 38 and 50. Understanding this step is crucial as it sets the stage for reducing the fraction.

• To find the GCD of 38 and 50, you start by listing their factors: - Factors of 38: 1, 2, 19, 38 - Factors of 50: 1, 2, 5, 10, 25, 50
• The common factors are the numbers present in both factor lists. In this case, they are 1 and 2.
• The greatest common factor of these is 2, so the GCD is 2.

Finding the GCD makes it possible to simplify the fraction by dividing both the numerator and the denominator by this GCD.

Lowest Terms

Fractions in the simplest form are said to be expressed in their lowest terms. This means that the greatest common divisor of the numerator and the denominator is 1.
Here's why it's important:

• Simplifying fractions to their lowest terms makes them easier to understand and compare.
• Mathematically, a fraction in its lowest terms is the most reduced form of that fraction.

For the fraction \( \frac{38}{50} \), once divided by the GCD (which is 2), you obtain \( \frac{19}{25} \). Since 19 and 25 share no common factors other than 1, this is the reduced form of the fraction. It cannot be simplified further, indicating it's already in its lowest terms.

Simplification of Fractions

Simplifying fractions involves reducing them to their lowest terms by eliminating common factors of the numerator and the denominator. This process emphasizes understanding and manipulating numbers logically.

Here's the step-by-step process to simplify a fraction:

• Identify the GCD: As we've seen, the first step is always to find the GCD of the given numbers.
• Divide by the GCD: Proceed by dividing both the numerator and the denominator by this divisor. For \( \frac{38}{50} \), dividing both numbers by 2 gives \( \frac{19}{25} \).
• Final Check: Ensure that the resulting fraction cannot be further reduced. This is done by confirming that the numerator and denominator share no common factors other than 1.

This method not only streamlines calculations but also aids in developing a deeper grasp of number theory, making problems more manageable.