Problem 8 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{-3}{15}$$

Step-by-Step Solution

Answer

The fraction \(\frac{-3}{15}\) reduces to \(\frac{-1}{5}\).

Step 1: Identify the GCD

To simplify the fraction \(\frac{-3}{15}\), we first need to find the greatest common divisor (GCD) of the numerator (-3) and the denominator (15). The GCD is the largest positive integer that divides both numbers without leaving a remainder. For -3 and 15, the GCD is 3.

Step 2: Divide both the Numerator and Denominator

Now that we have the GCD (which is 3), divide both the numerator and the denominator by the GCD. So, the term becomes: \(\frac{-3 \div 3}{15 \div 3} = \frac{-1}{5}\).

Step 3: Simplified Fraction

The fraction \(\frac{-1}{5}\) is fully reduced because 1 and 5 have no common divisors other than 1, and the negative sign remains in the numerator.

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a crucial concept when it comes to simplifying fractions. It refers to the largest positive integer that divides two or more numbers without leaving any remainder. To simplify a fraction like \(\frac{-3}{15}\), you first find the GCD of the numerator and denominator. Finding the GCD:

• List the positive factors of each number.
• Identify the highest number that appears in both lists.

For example, the factors of \(-3\) would be the same as \(3\) and include \(1\) and \(3\). The factors of \(15\) are \(1, 3, 5,\) and \(15\). Therefore, \(3\) is the highest common factor, making it the GCD of \(-3\) and \(15\). Calculating the GCD helps you know what number to divide both the numerator and the denominator by to simplify the fraction.

Simplifying Fractions

Simplifying fractions is all about making them as simple as possible while keeping their value the same. Once you've found the GCD, you can reduce the fraction by dividing both the numerator and the denominator by this number.Steps to Simplify:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• Write the simplified fraction.

For \(\frac{-3}{15}\), after you find that the GCD is \(3\), you divide both \(-3\) and \(15\) by \(3\). This results in the simplified fraction \(\frac{-1}{5}\). Simplification helps in making fractions easier to read and work with, and the process ensures that you arrive at the smallest possible equivalent fraction.

Numerator and Denominator

Every fraction consists of two parts: the numerator and the denominator. Understanding these components is essential to master working with fractions.Numerator:

• The top number in a fraction.
• Indicates how many parts of a whole are being considered.

Denominator:

• The bottom number in a fraction.
• Represents the total number of equal parts the whole is divided into.

In \(\frac{-3}{15}\), the numerator is \(-3\) and the denominator is \(15\). When simplifying, you analyze these two numbers to see how they can be divided by a common factor. A key point is that the numerator tells us the count of parts we have, while the denominator tells us what each part represents. Simplifying doesn't change the value of the fraction; it makes it easier to interpret and use.