Problem 107 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{51}{12}$$

Step-by-Step Solution

Answer

\(\frac{51}{12}\) reduces to \(\frac{17}{4}\) in lowest terms.

Step 1: Identify the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor of the numerator and the denominator. The numerator is 51, and the denominator is 12. The factors of 51 are 1, 3, 17, and 51. The factors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common factor between these two numbers is 3.

Step 2: Divide the Numerator and Denominator by GCD

Now that we know the GCD is 3, we divide both the numerator and the denominator by this number to reduce the fraction. So, \( \frac{51}{12} \) becomes:\[ \frac{51 \div 3}{12 \div 3} = \frac{17}{4} \]

Step 3: Check the Fraction

Finally, check if the new fraction \( \frac{17}{4} \) is in its lowest terms. The factors of 17 are 1 and 17, and the factors of 4 are 1, 2, and 4. Since there are no common factors except 1, the fraction is indeed in its lowest terms.

Key Concepts

greatest common divisor

Finding the Greatest Common Divisor (GCD) is an essential step in reducing fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder. For instance, when we have the fraction \( \frac{51}{12} \), we look at the factors of each number to find their GCD.

• The factors of 51 are: 1, 3, 17, 51.
• The factors of 12 are: 1, 2, 3, 4, 6, 12.

The largest number common to both lists is 3, hence the GCD of 51 and 12 is 3.
This divisor is crucial since it helps us shrink the fraction to its simplest form by dividing both terms by this GCD.

numerator and denominator

A fraction consists of two parts: the numerator and the denominator. The numerator is the top number that signifies how many parts we have, while the denominator is the bottom number that indicates into how many parts the whole is divided.
In the fraction \( \frac{51}{12} \), 51 is the numerator and 12 is the denominator. When reducing fractions, observing these parts for common factors such as the GCD becomes important.
To simplify, we divide both parts by the GCD. For example, we divided both 51 and 12 by their GCD, which is 3, to get \( \frac{17}{4} \). This process maintains the value of the fraction while expressing it in a simpler form.

lowest terms

A fraction is in its lowest terms when the only common divisor between the numerator and the denominator is 1. This means that the fraction cannot be reduced any further.
After finding the GCD and dividing both the numerator and the denominator by it, we check to ensure that they have no common factors left.
Taking the fraction \( \frac{51}{12} \) and reducing it using the GCD, we obtained \( \frac{17}{4} \). The factors of 17 are 1 and 17, and the factors of 4 are 1, 2, and 4. Recognizing that there is no common factor except 1 confirms that \( \frac{17}{4} \) is indeed in its lowest terms.
Reducing fractions to their lowest terms is a simplified way of expressing them, making calculations and comparisons easier.