Problem 5 - Algebra for College Students

Question

For Problems 1-8, express each rational number in reduced form. \(\frac{24}{-60}\)

Step-by-Step Solution

Answer

The reduced form is \(-\frac{2}{5}\).

Step 1: Understand What Reduced Form Means

A rational number in reduced form means that the numerator and the denominator have no common factors aside from 1. This is also called simplifying a fraction.

Step 2: Write Down the Fraction

The fraction given in the exercise is \( \frac{24}{-60} \). The negative sign can be associated with either the numerator or the denominator.

Step 3: Identify Common Factors

Find the greatest common factor (GCF) of the absolute values of the numerator 24 and the denominator 60. We list their factors to find the GCF. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24; factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The GCF is 12.

Step 4: Simplify the Fraction

Divide both the numerator and the denominator by their GCF, which is 12. \( \frac{24 \div 12}{-60 \div 12} = \frac{2}{-5} \).

Step 5: Adjust the Negative Sign

It is conventional to write the negative sign in front of the fraction. So, \( \frac{2}{-5} = -\frac{2}{5} \).

Step 6: Verify Simplification

Check that 2 and 5 have no common factors other than 1, confirming that \( -\frac{2}{5} \) is in its simplest form.

Key Concepts

Greatest Common Factor

When working with fractions, the greatest common factor (GCF) plays an essential role in simplifying them. The GCF of two numbers is the largest number that divides both numbers without leaving a remainder. In the context of fractions, finding the GCF helps us reduce fractions to their simplest form.

To find the GCF of 24 and 60, list out the factors of each number. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. For 60, they are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The largest number that appears in both lists is 12, making it the GCF. Using the GCF, we divide both the numerator and the denominator of the fraction by this number to simplify it. This technique is crucial in ensuring that fractions are expressed in their simplest form, with no factors left in common beyond 1.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. They are a vital part of the number system and include numbers like \(-\frac{2}{5}\), which we derived from the original fraction \(\frac{24}{-60}\).

In mathematics, rational numbers allow for a precise representation of numbers that aren't whole numbers. What makes these numbers particularly interesting is their properties and how they interact with operations like addition, subtraction, multiplication, and division. Moreover, any integer can also be considered a rational number, as it can be written as a fraction with a denominator of 1.

Understanding rational numbers is key to grasping concepts in fractions, ratios, and proportional reasoning, which are integral parts of algebra and higher-level mathematics.

Reducing Fractions

Reducing fractions, also known as simplifying fractions, involves making the fraction as simple as possible by ensuring that the numerator and the denominator are as small as possible while maintaining the same value. This process is important in simplifying mathematical expressions and making calculations easier.

To reduce a fraction, follow these steps:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCF.
• Adjust any negative signs if necessary to follow conventional fraction placement.

For example, simplify \(\frac{24}{-60}\) by dividing each part by the GCF, 12. This results in \(-\frac{2}{5}\), which is already in its simplest form.

Reducing fractions not only simplifies calculations but also provides a clearer understanding of the relative magnitudes of the numerator and denominator, enhancing comprehension of the underlying mathematical relationships.