Problem 7 - Algebra for College Students

Question

For Problems 1-8, express each rational number in reduced form. \(\frac{-16}{-56}\)

Step-by-Step Solution

Answer

The fraction simplifies to \( \frac{2}{7} \).

Step 1: Simplifying the Sign

The given fraction is \( \frac{-16}{-56} \). Both the numerator and the denominator have a negative sign, which makes the fraction positive. So, we simplify the sign and get \( \frac{16}{56} \).

Step 2: Finding the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{16}{56} \), we need to find the greatest common divisor of 16 and 56. First, list the factors: \(16 = 1, 2, 4, 8, 16\) and \(56 = 1, 2, 4, 7, 8, 14, 28, 56\). The common factors are 1, 2, 4, and 8. The greatest is 8.

Step 3: Reducing the Fraction

Divide both the numerator and the denominator by their GCD, 8. Thus, \( \frac{16}{8} = 2 \) and \( \frac{56}{8} = 7 \). Therefore, \( \frac{16}{56} \) reduces to \( \frac{2}{7} \).

Step 4: Conclusion

The fraction \( \frac{-16}{-56} \) simplifies to \( \frac{2}{7} \) in reduced form.

Key Concepts

Understanding Rational Numbers

Rational numbers are an essential concept in mathematics and appear often. A rational number refers to a number that can be expressed as a fraction, where both the numerator and the denominator are integers. Importantly, the denominator should not be zero because division by zero is undefined.
Examples of rational numbers include fractions like \( \frac{3}{4} \), \( \frac{-5}{2} \), or even whole numbers like 7 (which can be expressed as \( \frac{7}{1} \)). Rational numbers can also be positive or negative.

• Negative rational numbers have either the numerator or the denominator negative, but not both simultaneously.
• Positive rational numbers either have both parts positive or both negative. A fraction like \( \frac{-2}{-5} \) is positive, as the negatives cancel each other out.

Finding the Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest integer that divides two or more numbers without leaving a remainder. When simplifying fractions, finding the GCD helps to reduce the fraction to its simplest form by evenly dividing both the numerator and the denominator.
To find the GCD:

• First, list the factors of each number. Factors are numbers that can divide the number without leaving a remainder. For example, the factors of 16 are 1, 2, 4, 8, and 16.
• Identify the common factors shared by both the numerator and the denominator. For instance, for 16 and 56, the common factors are 1, 2, 4, and 8.
• The largest of these common factors, in this case, 8, is the GCD.

Recognizing and applying the GCD is crucial in making fractions simpler and more manageable.

Simplification Steps of Fractions

Simplifying fractions involves reducing them to their simplest form. The process makes numbers easier to work with, especially in operations like addition or multiplication. Here's a breakdown of the simplification process:

• Simplify signs first: If both the numerator and the denominator are negative, the fraction becomes positive. Just remove the negative signs.
• Find the GCD: Locate the greatest common divisor of the numerator and the denominator. As explained, this will involve listing and comparing factors.
• Divide by the GCD: Use the GCD to divide both the numerator and the denominator. This step reduces the fraction to its simplest form. For \( \frac{16}{56} \), divide both by 8 to get \( \frac{2}{7} \).

A simplified fraction like \( \frac{2}{7} \) is easier to interpret and use in calculations, making this step invaluable for efficiency and clarity in math.