Problem 100 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{30}{105}$$

Step-by-Step Solution

Answer

The fraction \( \frac{30}{105} \) reduces to \( \frac{2}{7} \).

Step 1: Find the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{30}{105} \) to its lowest terms, we first need to find the greatest common divisor (GCD) of 30 and 105. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105. The common factors between them are 1, 3, 5, and 15. Therefore, the GCD is 15.

Step 2: Divide the Numerator and Denominator by the GCD

Using the GCD found in Step 1, which is 15, divide both the numerator and the denominator of the fraction \( \frac{30}{105} \) by 15. Thus, \( \frac{30}{105} = \frac{30 \div 15}{105 \div 15} = \frac{2}{7} \)

Step 3: Verify the Reduced Fraction

To verify, we check if 2 and 7 have any common divisors other than 1. The factors of 2 are 1 and 2, and the factors of 7 are 1 and 7. Since the only common factor is 1, \( \frac{2}{7} \) is in its lowest terms.

Key Concepts

Greatest Common Divisor

In mathematics, finding the greatest common divisor (GCD) is crucial for simplifying fractions. The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder.
For example, when reducing the fraction \( \frac{30}{105} \), you determine the GCD of the numerator (30) and the denominator (105).

• List the factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30.
• List the factors of 105: 1, 3, 5, 7, 15, 21, 35, and 105.

The common factors are 1, 3, 5, and 15, making 15 the greatest. Once you find the GCD, it simplifies further calculations.

Numerator and Denominator

The numerator and denominator are fundamental components of any fraction. The numerator is the top number, which tells how many parts you have. The denominator, on the bottom, tells how many equal parts make a whole.
In the fraction \( \frac{30}{105} \), 30 is the numerator and 105 is the denominator.

• Numerator (30): Indicates the number of parts being considered.
• Denominator (105): Represents the total number of equal parts.

Understanding these components is necessary to properly reduce fractions and interpret what part of a whole they represent.

Reducing Fractions

Reducing fractions, also known as simplifying, involves making the fraction as simple as possible while maintaining its value. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
To reduce \( \frac{30}{105} \):

• Divide 30 (numerator) by the GCD of 15 to get 2.
• Divide 105 (denominator) by 15 to get 7.

Thus, \( \frac{30}{105} \) reduces to \( \frac{2}{7} \). The resulting fraction represents the same value in more straightforward terms, which are easier to grasp.

Lowest Terms

A fraction is in its lowest terms when the numerator and denominator have no common divisor other than 1. This means the fraction is as simplified as it can be without changing its value.
After reducing \( \frac{30}{105} \) using its GCD, you arrive at \( \frac{2}{7} \).
Checking the new numerator and denominator, 2 and 7, shows:

• Factors of 2: 1, 2.
• Factors of 7: 1, 7.

Since they have no common factors other than 1, \( \frac{2}{7} \) is indeed in its lowest terms, verifying it's simplified correctly.