Problem 26 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{8}{40} $$

Step-by-Step Solution

Answer

The fraction \( \frac{8}{40} \) simplifies to \( \frac{1}{5} \).

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor of the numerator and the denominator. For the fraction \( \frac{8}{40} \), list the factors of 8 (which are 1, 2, 4, 8) and the factors of 40 (which are 1, 2, 4, 5, 8, 10, 20, 40). The greatest common factor is 8.

Step 2: Divide Numerator and Denominator by GCF

Divide both the numerator and the denominator by the greatest common factor. Begin by dividing 8 by 8, which equals 1, and 40 by 8, which equals 5. This simplifies the fraction from \( \frac{8}{40} \) to \( \frac{1}{5} \).

Step 3: Verify the Simplification

Ensure the fraction cannot be reduced further. Since 1 and 5 have no common factors other than 1, the fraction \( \frac{1}{5} \) is in its simplest form.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept when simplifying fractions. The GCF of two numbers is the largest number that divides both of them without leaving a remainder. To find it, you list all the factors of each number and choose the greatest one they have in common.

• Factors of 8: 1, 2, 4, 8
• Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

For instance, the GCF of 8 and 40 is 8, because it is the largest number that appears in both lists. This step is crucial when reducing fractions, as you need the GCF to properly simplify the fraction to its lowest terms.

Numerator and Denominator

In a fraction, the numerator is the number above the line, while the denominator is the number below the line. These parts of the fraction work together to show a ratio of how many parts we have (numerator) out of a whole (denominator).
For example, in the fraction \( \frac{8}{40} \), 8 is the numerator and 40 is the denominator.

• The numerator (8) tells us the number of equal parts we are considering.
• The denominator (40) tells us the total number of these equal parts.

When simplifying fractions, both the numerator and the denominator need to be divided by the greatest common factor to find a smaller, equivalent fraction.

Reducing Fractions

Reducing fractions means simplifying them to their lowest terms, where the numerator and denominator have no common factors other than 1. The process starts with finding the GCF, then dividing both the numerator and the denominator by this number. Let's take the fraction \( \frac{8}{40} \) for example:1. Identify the GCF, which is 8.2. Divide both the numerator and the denominator by 8: - \( 8 \div 8 = 1 \) - \( 40 \div 8 = 5 \)3. Resulting in the simplified fraction: \( \frac{1}{5} \).
This fraction is fully reduced because no number other than 1 can divide both 1 and 5. Reducing fractions makes them easier to work with and understand when comparing ratios or solving mathematical problems.