Problem 10 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{9}{-51}$$

Step-by-Step Solution

Answer

The fraction reduced to lowest terms is \( \frac{-3}{17} \).

Step 1: Understand the Problem

We need to reduce the fraction \( \frac{9}{-51} \) to its lowest terms by finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that GCD.

Step 2: Identify the Numerator and Denominator

The fraction is \( \frac{9}{-51} \). Here, \(9\) is the numerator and \(-51\) is the denominator.

Step 3: Find the GCD of Numerator and Denominator

To reduce the fraction, we need to find the GCD of \(9\) and \(51\). The numbers that divide both 9 and 51 are 1 and 3. Hence, the GCD is 3.

Step 4: Divide by the GCD

Divide both the numerator and the denominator by the GCD: \( \frac{9 \div 3}{-51 \div 3} = \frac{3}{-17} \).

Step 5: Adjust the Sign

A fraction with a negative denominator can be adjusted to have a positive denominator by moving the negative sign to the numerator. Hence, \( \frac{3}{-17} \) becomes \( \frac{-3}{17} \).

Key Concepts

Greatest Common Divisor

The concept of the greatest common divisor (GCD) is vital in simplifying fractions. The GCD of two numbers is the largest number that perfectly divides both without leaving a remainder. For example, when reducing the fraction \( \frac{9}{-51} \), we need the GCD of 9 and 51. This step ensures that our simplified fraction is in its simplest form.

• To find the GCD, list the factors of each number.
• For 9: 1, 3, and 9.
• For 51: 1, 3, 17, and 51.
• The largest common factor is 3, making it the GCD.

With the GCD identified, you proceed by dividing both the numerator and denominator by this value. This step turns \( \frac{9}{-51} \) into \( \frac{3}{-17} \). Remember, finding the GCD is fundamental in reducing fractions effectively, ensuring they remain equivalent but simpler.

Numerator and Denominator

Every fraction has two essential components: the numerator and the denominator. In \( \frac{9}{-51} \), 9 is the numerator, and -51 is the denominator. Understanding these terms is crucial for manipulating fractions correctly.

• The numerator represents the number of parts we currently have.
• The denominator signifies the total number of equal parts something is divided into.

In our example fraction \( \frac{9}{-51} \), the numerator 9 shows how many parts you have, while the -51 in the denominator indicates the total number of parts those 9 units are divided from. Knowing how to identify and work with these numbers ensures you reduce fractions properly, maintaining their original value but in a simpler form.

Lowest Terms

Reducing a fraction to its lowest terms means making it as simple as possible. The aim is to have a numerator and denominator that share no common factors other than one.

• This means, once the GCD is found, you use it to divide both the numerator and denominator.
• For example, 9 and -51 share a GCD of 3.
• Divide them by 3 to obtain \( \frac{3}{-17} \).

Once reduced, if the denominator is negative, you might want to adjust the sign by either moving it to the numerator or making it positive, like so: \( \frac{-3}{17} \). Reducing to lowest terms speeds up arithmetic with fractions and presents them in a clear, concise manner. This ensures the fraction is minimal and easy to work with without altering its value.