Problem 44 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$ \frac{11}{12}-\frac{3}{12} $$

Step-by-Step Solution

Answer

The difference is \( \frac{2}{3} \).

Step 1: Identify and Interpret the Problem

We start by noting the problem: We need to find the difference of two fractions, \( \frac{11}{12} \) and \( \frac{3}{12} \). Both fractions have the same denominator, making it a straightforward subtraction problem.

Step 2: Subtract the Numerators

Since the denominators are the same, \(12\), we can subtract the numerators directly: \(11 - 3 = 8 \). The resultant fraction is \( \frac{8}{12} \).

Step 3: Simplify the Fraction

Now simplify \( \frac{8}{12} \). Identify the greatest common factor (GCF) between 8 and 12, which is 4. Divide both the numerator and the denominator by 4: \( \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \).

Step 4: Verify the Simplified Form

Check if \( \frac{2}{3} \) can be simplified further. Since 2 and 3 have no common factors other than 1, \( \frac{2}{3} \) is in its simplest form.

Key Concepts

Simplifying Fractions

Whenever you perform operations with fractions, it often involves simplifying the result. Simplifying a fraction means rewriting it so that the numerator and the denominator have no common factors other than 1. This process makes fractions easier to read and compare.

• To start simplifying, find the Greatest Common Factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.
• The result is a simpler, or simplest equivalent fraction.

For example, with the fraction \(\frac{8}{12}\), the GCF is 4. Dividing both by 4, we get \(\frac{2}{3}\). Always double-check your work to ensure no further simplifying can be done. If the numerator and denominator have no common factors other than 1, you've done it right! This process is crucial for clarity and for preparing fractions for further arithmetic operations.

Common Denominator

In fraction subtraction, like many fraction operations, a common denominator is a key component. A common denominator is simply a shared divisor that the individual fractions can all use. Having a common denominator allows for direct addition or subtraction of fractions.

• You equate the denominators of the fractions involved in the operation.
• This might mean adjusting the numerators to ensure the fractions stay equivalent.
• With this common baseline, focus on the numerators alone for direct computation.

In the exercise \(\frac{11}{12} - \frac{3}{12}\), both fractions already share the denominator 12. This makes subtraction straightforward, as you manage only the numerators, resulting in the new fraction \(\frac{8}{12}\). Starting from a common denominator simplifies operations significantly, which helps avoid any unnecessary complexity.

Greatest Common Factor (GCF)

Understanding the Greatest Common Factor (GCF) is crucial in working with and simplifying fractions. The GCF is the largest integer that can evenly divide both the numerator and the denominator of a fraction.

• Use the GCF to reduce fractions to their simplest form.
• Identifying the GCF involves determining all divisors of the numerator and the denominator, and selecting the highest common one.
• Apply this factor by dividing both the top and bottom of the fraction to simplify it.

For instance, with \(\frac{8}{12}\), you look for the GCF of 8 and 12, which is 4. Dividing both terms by 4 simplifies the fraction to \(\frac{2}{3}\). Without this factor, simplifying fractions would be much harder, possibly leading to mistakes. Mastery of finding the GCF simplifies not just fractions, but many areas of math including algebra and number theory.