Problem 109 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{51}{54}$$

Step-by-Step Solution

Answer

The fraction \(\frac{51}{54}\) reduces to \(\frac{17}{18}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce the fraction to its lowest terms, we need to find the greatest common divisor of the numerator and the denominator. The numerator is 51 and the denominator is 54. To find the GCD of 51 and 54, list the factors of both numbers: \(51: 1, 3, 17, 51\) and \(54: 1, 2, 3, 6, 9, 18, 27, 54\). The greatest common factor is 3.

Step 2: Divide the Numerator and Denominator by the GCD

Now that we have the GCD, which is 3, divide both the numerator (51) and the denominator (54) by 3. Calculating \(\frac{51}{3}\) gives us 17. Calculating \(\frac{54}{3}\) gives us 18.

Step 3: Write the Fraction in Lowest Terms

Since both the numerator and the denominator were divided by their greatest common divisor, you write the reduced fraction as \(\frac{17}{18}\). This is the fraction in its lowest terms since 17 and 18 have no common factors other than 1.

Key Concepts

Greatest Common Divisor

To reduce a fraction, one of the key steps is to find the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder.

• To determine the GCD, list all of the factors for both the numerator and the denominator.
• Factors are numbers that divide into another number completely, without leaving any leftovers.

For instance, with the fraction \(\frac{51}{54}\), the factors of 51 are 1, 3, 17, and 51. For 54, they are 1, 2, 3, 6, 9, 18, 27, and 54. The largest common factor is 3, making it the GCD.
Finding the GCD is crucial because once you've found it, you can divide both the numerator and the denominator by this number to help simplify the fraction.

Numerator and Denominator

In a fraction, understanding the roles of the numerator and the denominator is essential. The numerator is the top number. It represents how many parts of a whole are being considered. The denominator, on the other hand, is the bottom number and tells us into how many equal parts the whole is divided.

• Take for example the fraction \(\frac{51}{54}\): here, 51 is the numerator, and 54 is the denominator.
• The fraction \(\frac{51}{54}\) expresses that you have 51 parts out of a whole that is divided into 54 equal parts.

When simplifying fractions, you adjust both the numerator and the denominator. In this case, after finding the GCD, you divide the numerator and denominator by the GCD, reducing the size of the fraction without changing its overall value.

Lowest Terms

A fraction is said to be in "lowest terms" when the numerator and denominator share no common factors other than 1. This means that the fraction cannot be simplified further.

• The process of reducing a fraction to its lowest terms involves dividing both the numerator and the denominator by their GCD.
• For \(\frac{51}{54}\), after dividing both 51 and 54 by their GCD of 3, you get \(\frac{17}{18}\).

Here, 17 and 18 don't share any other common factors, meaning this fraction is in its simplest form.
Keeping fractions in their lowest terms is helpful for making calculations easier and recognizing equivalent fractions quickly.