Problem 4 - Algebra for College Students

Question

For Problems 1-8, express each rational number in reduced form. \(\frac{-14}{42}\)

Step-by-Step Solution

Answer

The reduced form is \(-\frac{1}{3}\).

Step 1: Find the Greatest Common Divisor (GCD)

To reduce a fraction, we need to divide the numerator and denominator by their greatest common divisor (GCD). The GCD of 14 and 42 is 14.

Step 2: Divide Both Numerator and Denominator by GCD

Divide the numerator (-14) and the denominator (42) by their GCD, which is 14, to simplify the fraction: \(-\frac{14}{42} = -\frac{14 \div 14}{42 \div 14} = -\frac{1}{3}\).

Step 3: Write the Simplified Fraction

The reduced form of the fraction \(-\frac{14}{42}\) is \(-\frac{1}{3}\).

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is a fundamental concept in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD ensures that we can reduce the fraction to its simplest form. In our example, the GCD of 14 and 42 is 14. This means 14 is the highest number that both 14 and 42 can be divided by evenly.

• The GCD is essential for fraction reduction as it helps find the simplest equivalent fraction.
• Determining the GCD usually involves listing the factors of the numbers and identifying the greatest number that both lists have in common.
• Euclidean algorithm is another efficient method to find the GCD, especially for larger numbers.

Fraction Reduction

Fraction reduction streamlines a fraction to its simplest form using the GCD of its numerator and denominator. This process does not change the value of the fraction; it merely simplifies the expression. By reducing fractions, we handle numbers that are easier to interpret and operate on.

• A fraction is in its reduced form when its numerator and denominator have no common factors other than 1.
• It helps in comparing fractions, computing arithmetic operations, and understanding the size or equivalence of the fractions involved.

For instance, the fraction \( \frac{-14}{42} \) simplifies to \(-\frac{1}{3}\). Both share the GCD 14, reducing complexity without altering the fraction's value.

Numerator and Denominator Division

After determining the GCD, simplifying a fraction involves dividing both the numerator and the denominator by this number. This division helps achieve the reduced form of the fraction and is a straightforward calculation.

• This process ensures that both components of the fraction are proportionally simplified.
• It is essential to divide accurately to maintain the numerical integrity of the fraction.

In our example, dividing both \(-14\) and \(42\) by their GCD (14) gives us \(-1\) and \(3\) respectively. So, \(\frac{-14}{42}\) simplifies to \(-\frac{1}{3}\). Dividing consistently on both parts is crucial for accurately simplifying fractions.