Problem 62 - Elementary Algebra

Question

Divide and reduce to lowest terms. $$ (-25) \div 153 $$

Step-by-Step Solution

Answer

The simplified division of (-25) by 153 is \(-\frac{25}{153}\).

Step 1: Understand the Division Problem

The problem asks for the division of (-25) by 153. This means we need to determine how many times 153 fits into (-25) or what fraction or ratio represents this division.

Step 2: Perform the Division

Divide (-25) by 153. In fraction terms, this is written as \(-\frac{25}{153}\). This is a negative fraction because we are dividing a negative number by a positive number.

Step 3: Simplify the Fraction

To reduce the fraction to its lowest terms, find the greatest common divisor (GCD) of 25 and 153. The GCD of 25 and 153 is 1, as they have no common factors other than 1.

Step 4: Conclude the Simplification

Since the GCD is 1, \(-\frac{25}{153}\) is already in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

When we talk about fraction simplification, the Greatest Common Divisor (GCD) plays a crucial role. The GCD is the largest number that can evenly divide two given numbers without leaving a remainder. To find the GCD, you can list all the factors of each number and then identify the largest one they share. For example, the factors of 25 are 1, 5, and 25. The factors for 153 are 1, 3, 9, 17, 51, and 153. Here, the only common factor is 1.
This means that the greatest common divisor is 1, indicating that the given fraction, \(-\frac{25}{153}\), is already fully simplified because there are no larger common factors to reduce it further. Remember, finding the GCD is essential in determining how much a fraction can be simplified.

Simplifying Fractions

Simplifying a fraction means to express it in its simplest form. A fraction is simplified when the numerator and the denominator have no common factors other than 1. To simplify fractions, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

For instance, in the case of \(-\frac{25}{153}\), the GCD is 1. Therefore, we cannot reduce it further because \(-25\) and \153\ have no other common divisors. Simplifying fractions helps in making calculations easier and keeps results neat.

Negative Fractions

Negative fractions may look confusing, but they follow the same basic rules as their positive counterparts. A fraction is called negative if its numerator or denominator, but not both, is negative. With \(-\frac{25}{153}\), the negative sign shows the overall value of the fraction is less than zero.
It's important to note where the negative sign is placed. In mathematics, the negative sign can appear either in front of the entire fraction, in the numerator, or in the denominator, but it typically simplifies to a more common format. So, \(-\frac{25}{153}\) is considered the same as \frac{-25}{153}\.
When performing operations like addition or multiplication with negative fractions, remember that the rules of negatives come into play. The sign of the result depends on the signs of the numbers you are working with. Keeping these rules in mind ensures you handle negative fractions correctly.