Problem 8 - Algebra for College Students

Question

For Problems 1-8, express each rational number in reduced form. \(\frac{-30}{-42}\)

Step-by-Step Solution

Answer

The rational number \( \frac{-30}{-42} \) reduces to \( \frac{5}{7} \).

Step 1: Identify the Rational Number

The given rational number is \( \frac{-30}{-42} \). We need to simplify this rational number by reducing it to its lowest terms.

Step 2: Determine the Common Factors

Calculate the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 30 and 42 is 6.

Step 3: Simplify the Fraction

Divide both the numerator and the denominator by their GCD. So, \( \frac{-30}{-42} = \frac{-30/6}{-42/6} = \frac{-5}{-7} \).

Step 4: Simplify the Signs

Since both the numerator and the denominator are negative, the rational number becomes positive. Therefore, \( \frac{-5}{-7} = \frac{5}{7} \).

Step 5: Final Reduced Form

The rational number simplified is \( \frac{5}{7} \). This is the final reduced form.

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a key concept in mathematics. It is used to identify the largest number that divides two or more numbers without leaving a remainder. In the context of simplifying rational numbers, the GCD helps us reduce fractions to their simplest form. To find the GCD of two numbers, you list the factors of each number and identify the largest factor that appears in both lists.
For instance, to find the GCD of 30 and 42, start by determining their factors:

• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

The largest common factor here is 6, making it the GCD.
Understanding how to find the GCD is crucial for reducing fractions effectively.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their most basic form, where the numerator and denominator have no common factors other than 1. This involves dividing the top and bottom of the fraction by their greatest common divisor (GCD). Simplifying helps in understanding and comparing fractions more easily.
For example, consider the fraction \(\frac{-30}{-42}\). By identifying the GCD as 6, we divide both parts by this number:

• Numerator: \(-30 \div 6 = -5\)
• Denominator: \(-42 \div 6 = -7\)

The simplified form of the fraction is \(\frac{-5}{-7}\). By further simplifying the negative signs, we arrive at the final reduced form \(\frac{5}{7}\), which is easier to work with and understand in various mathematical contexts.

Positive and Negative Rational Numbers

Rational numbers can be either positive or negative, and the sign plays an essential role in understanding them. A rational number is positive if the signs of the numerator and the denominator are the same. It is negative if one is positive and the other is negative.
In our example of \(\frac{-30}{-42}\), both the numerator and the denominator are negative, leading to a positive rational number when simplified. This is because a negative divided by a negative equals a positive.Here's how it works:

• \(-5 \div -7\) gives \(5/7\)

Understanding the effect of signs is important for simplifying complex expressions and solving equations involving rational numbers.