Problem 115 - Fundamentals of Mathematics

Question

For the following problems, find the mistake. $$ \frac{3}{24}=\frac{\overline{)3}}{\overline{)33} \cdot 8}=\frac{0}{8}=0 $$

Step-by-Step Solution

Answer

The mistake is in incorrectly cancelling numbers and simplifying to zero, instead of \( \frac{1}{8} \).

Step 1: Understand the Expression

The original fraction is \( \frac{3}{24} \). We need to determine whether this fraction simplifies correctly.

Step 2: Simplify the Fraction Correctly

\( \frac{3}{24} \) should be simplified by finding the greatest common divisor (GCD) of 3 and 24, which is 3. Then divide both the numerator and the denominator by this GCD: \( \frac{3 \div 3}{24 \div 3} = \frac{1}{8} \).

Step 3: Identify the Mistake

The mistake occurs when crossing out 3 in both the numerator and the one in 33, resulting in a denominator \( \overline{)33} \cdot 8 = \overline{)3}8 \), which incorrectly simplifies to zero.

Step 4: Confirm Correct Simplification

Since \( \frac{3}{24} \) simplifies correctly to \( \frac{1}{8} \), the given solution which concludes to zero is incorrect. The process erroneously interprets the number by trying to cancel non-existing factors.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in fraction simplification. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. In our example, we are simplifying the fraction \( \frac{3}{24} \). To do this correctly, we first need to determine the GCD of the numerator, 3, and the denominator, 24, which is 3.

• Step 1: List the factors of 3: 1, 3.
• Step 2: List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
• Step 3: The common factors are 1 and 3. Thus, the GCD is 3.

Finding such common factors allows us to reduce fractions to their simplest form by dividing both the numerator and the denominator by the GCD. This crucial step ensures the fraction is expressed in the simplest terms possible without altering its value.

Numerator and Denominator

Fractions consist of two essential parts: the numerator and the denominator. The numerator is the number above the fraction line, while the denominator is the number below it. These parts play a critical role in the simplification process. In the fraction \( \frac{3}{24} \), 3 is the numerator and 24 is the denominator.
Understanding each part's function is crucial:

• The numerator indicates how many parts of a whole are being considered.
• The denominator shows the total number of equal parts that make up the whole.

When simplifying fractions, it's important to maintain the relationship between these two parts by dividing both by their common factors. This step ensures that their ratio remains the same, preserving the original value of the fraction even after simplification.

Mathematical Error Identification

Mathematical error identification is an impactful skill, especially when simplifying fractions. Let's analyze the error in the original fraction simplification \( \frac{3}{24} = \frac{\overline{)3}}{\overline{)33} \cdot 8} = \frac{0}{8} = 0 \).
The error stems from an incorrect process of canceling out the numerator with a numeric component that doesn't correctly exist in the denominator (i.e., transforming \( 33 \cdot 8 \) incorrectly).

• First mistake: Canceling the numerator 3 with a misrepresented 3 in a bogus 33 in the denominator.
• Second mistake: Erroneous conclusion of cancelling to zero, which transforms the fraction incorrectly to zero.

Understanding such mistakes will help avoid them. Always focus on accurate factorization and common factors without assuming relationships that do not exist.

Correct Simplification Techniques

Correct simplification of fractions involves a step-by-step approach that ensures accuracy and clarity. Here is how one can approach simplifying \( \frac{3}{24} \) correctly.
The simplification process:

• Step 1: Identify the greatest common divisor (GCD) of the numerator and denominator. For 3 and 24, the GCD is 3.
• Step 2: Divide both the numerator and the denominator by the GCD:
  \[\frac{3 \div 3}{24 \div 3} = \frac{1}{8}\]
• Step 3: Recheck your simplification to ensure no steps were missed and calculations are accurate.
• Step 4: Confirm that the simplified fraction is in its lowest terms by ensuring no further common divisors exist.

Following these precise steps not only ensures the fraction is simplified correctly but also bolsters conceptual understanding of fraction manipulation. Practice these techniques will help prevent errors and build confidence in handling math problems.