Problem 11 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{-54}{-56}$$

Step-by-Step Solution

Answer

The fraction \( \frac{-54}{-56} \) simplifies to \( \frac{27}{28} \).

Step 1: Simplify the Double Negative

The given fraction is \( \frac{-54}{-56} \). Since both the numerator and denominator are negative, the fraction simplifies to a positive fraction. Therefore, this is equivalent to \( \frac{54}{56} \).

Step 2: Find the Greatest Common Divisor (GCD)

To simplify the fraction, we need to find the greatest common divisor of the numerator and the denominator, which are 54 and 56. List the factors: - Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54 - Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 The common factors are 1 and 2. Therefore, the GCD is 2.

Step 3: Simplify the Fraction

Now that we have the GCD, simplify the fraction \( \frac{54}{56} \) by dividing both the numerator and the denominator by the GCD, which is 2:\[ \frac{54 \div 2}{56 \div 2} = \frac{27}{28} \]

Step 4: Verify the Simplest Form

Check if \( \frac{27}{28} \) can be simplified further. Since 27 and 28 have no common factors other than 1, \( \frac{27}{28} \) is in its simplest form.

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a fundamental concept in simplifying fractions. It is the largest integer that divides two numbers without leaving a remainder. Finding the GCD helps us to reduce fractions efficiently by ensuring no common factors remain between the numerator and the denominator other than 1.
To find the GCD of two numbers, like 54 and 56 in this exercise, we first list all their factors:

• Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
• Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56

Next, we identify the common factors. For 54 and 56, the common factors are 1 and 2, so the greatest common divisor is 2. This GCD is then used to simplify the fraction effectively.

Simplifying Fractions

Simplifying fractions is all about reducing them to their simplest form. This means expressing the fraction with the smallest possible whole numbers while keeping the same value. To accomplish this, you divide both the numerator and the denominator by their GCD.
Let's consider the fraction from the exercise: \( \frac{54}{56} \). We already found that their GCD is 2. By dividing both the top and bottom of the fraction by 2, we simplify it:

• \(54 \div 2 = 27\)
• \(56 \div 2 = 28\)

Thus, \( \frac{54}{56} \) simplifies to \( \frac{27}{28} \). Always check if the new fraction can be simplified further. If the numerator and the denominator have no more common factors aside from 1, then the fraction is in its simplest form.

Arithmetic Operations

Arithmetic operations form the building blocks for managing and simplifying fractions. Key operations include addition, subtraction, multiplication, and division. Simplifying a fraction primarily involves division, as seen in this exercise.
Here, we started with division to remove the negative signs. Then, we used division again to simplify the fraction by dividing both the numerator and the denominator by their GCD. It's important to grasp how division aids in reducing fractions by eliminating common factors, yielding a simpler equivalent form.
In more complex cases, knowing how to handle arithmetic operations with fractions can help mix or separate terms and solve equations that involve ratios.