Problem 58 - Algebra 1

Question

Simplify the fraction. $$\frac{-15}{125}$$

Step-by-Step Solution

Answer

The simplified form of \(-\frac{15}{125}\) is \(-\frac{3}{25}\).

Step 1: Identify the Numerator and Denominator

In this fraction \(-\frac{15}{125}\), -15 is the numerator and 125 is the denominator.

Step 2: Determine the Greatest Common Divisor (GCD)

The task is to look for the greatest number that can divide both -15 and 125. In this case, it is 5.

Step 3: Divide the Numerator and Denominator by the GCD

Divide both -15 and 125 by their GCD, which is 5, to get \(-\frac{15 \div 5}{125 \div 5} = -\frac{3}{25}\).

Step 4: Check Your Work

Now, multiply the reduced numbers 3 and 25 by the GCD (5) to ensure it gives back the original numerator and denominator. In this case, \((-3 * 5) = -15\) and \( (25 * 5) = 125 \), therefore, \(-\frac{3}{25} = -\frac{15}{125}\).

Key Concepts

Greatest Common Divisor

Understanding the Greatest Common Divisor (GCD) is essential for simplifying fractions efficiently. The GCD of two numbers is the largest number that can divide both numbers without leaving a remainder. For example, to simplify the fraction \(-\frac{15}{125}\), we first need to find the GCD of -15 and 125.
To do this, list the factors of each number:

• -15 has factors: -1, 1, -3, 3, -5, 5, -15, 15
• 125 has factors: 1, 5, 25, 125

From the above lists, the largest factor shared by both numbers is 5.
Identifying the GCD allows us to divide both the numerator and denominator by this number to simplify the fraction.

Numerators and Denominators

In a fraction, the top part is the numerator, and the bottom part is the denominator. These components are crucial for fraction operations. In the exercise given, the fraction \(-\frac{15}{125}\) has the following parts:

• Numerator: -15, representing the number of equal parts being considered or taken.
• Denominator: 125, representing the total number of equal parts in a whole.

Both of these parts are integral in understanding and performing operations like addition, subtraction, or simplification of fractions. A change in either affects the value of the fraction.
Whenever simplifying fractions, always begin by identifying the numerator and the denominator before proceeding with any calculations.

Fraction Reduction

Reducing fractions, also known as fraction simplification, involves simplifying a fraction to its lowest terms. This means finding an equivalent fraction which has the smallest possible numerator and denominator.
The process can be broken down into simple steps:1. **Identify the GCD:** As explained earlier, find the greatest common divisor of both the numerator and the denominator.2. **Divide Both by the GCD:** Divide the numerator and the denominator by their GCD to simplify the fraction.
For instance, with \(-\frac{15}{125}\), we identified the GCD as 5. By dividing both the numerator (-15) and the denominator (125) by 5, we get the reduced fraction \(-\frac{3}{25}\).
The aim is to make the fraction easier to work with without changing its value. Remember, a reduced fraction is equivalent to the original fraction.