Problem 12 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{-24}{-80}$$

Step-by-Step Solution

Answer

\( \frac{-24}{-80} \) simplifies to \( \frac{3}{10} \).

Step 1: Identify the Greatest Common Divisor (GCD)

First, identify the greatest common divisor (GCD) of the numerator and the denominator in the fraction \( \frac{-24}{-80} \). For \( 24 \) and \( 80 \), the GCD is \( 8 \). This means 8 is the largest number that divides both 24 and 80 without leaving a remainder.

Step 2: Simplify the Fraction Using GCD

Now, simplify the fraction by dividing both the numerator and the denominator by their GCD. Divide \( 24 \) by 8 to get \( 3 \), and divide \( 80 \) by 8 to get \( 10 \). This gives us the simplified fraction \( \frac{3}{10} \) since the negative signs cancel each other.

Step 3: Verify the Simplified Fraction

Finally, verify that \( \frac{3}{10} \) is in its simplest form by confirming that the numerator and the denominator have no common factors other than 1. Since 3 and 10 have no common factors other than 1, \( \frac{3}{10} \) is indeed in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

The concept of the Greatest Common Divisor (GCD) is vital when working with fractions, as it helps simplify them. The GCD of two numbers is the largest number that can evenly divide both without leaving a remainder. To find the GCD, you can use a few different methods.

• The **listing method**, where you list all the factors of both numbers and identify the largest common one.
• Another method you can use is the **Euclidean algorithm**, which involves division steps to identify the GCD.

For example, if you have the numbers 24 and 80, listing the factors of 24 gives you 1, 2, 3, 4, 6, 8, 12, and 24. Listing the factors of 80 gives you 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80. The largest number common to both lists is 8, hence the GCD is 8. Calculating the GCD is essential because it provides the necessary information to reduce fractions to their simplest form.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest possible form while maintaining the same value. This is achieved by dividing both the numerator and the denominator by their GCD.

• For instance, consider the fraction \( \frac{24}{80} \). The GCD of the numbers 24 and 80 is 8.
• By dividing the numerator (24) by 8, you get 3, and dividing the denominator (80) by 8 gives you 10.

The fraction is simplified to \( \frac{3}{10} \). It's like reducing both part and whole of the fraction equally, retaining the same proportion. This process is crucial because it makes fractions easier to understand and work with. After simplification, fractions are often less cumbersome to deal with in further calculations.

Fraction Verification

Once you've simplified a fraction, it’s important to verify that it's in the simplest form. This involves ensuring there are no common factors between the numerator and the denominator other than 1. Verification is a good practice because it confirms the success of your simplification process.

• For instance, after simplifying the fraction \( \frac{24}{80} \) to \( \frac{3}{10} \), check for common factors.
• Both 3 and 10 only share a common factor of 1, confirming that \( \frac{3}{10} \) is indeed fully simplified.

When fractions are verified correctly, you have confidence that they are in their most useful form for arithmetic operations, algebraic expressions, or comparison with other fractions. Verification is the last step to ensure that no further reduction is possible, which is especially important in complex calculations.