Problem 116 - Fundamentals of Mathematics

Question

For the following problems, find the mistake. $$ \frac{8}{10}=\frac{\overline{)2}+6}{\overline{) 2}+8}=\frac{6}{8}=\frac{3}{4} $$

Step-by-Step Solution

Answer

Mistake: Incorrect simplification; \(\frac{8}{10} = \frac{4}{5}\), not \(\frac{3}{4}\).

Step 1: Identify Fractions on Both Sides

The given problem is \( \frac{8}{10} = \frac{2 + 6}{2 + 8} \). We need to first ensure the fractions on both sides are equivalent and properly simplified. Let's check the simplification step-by-step.

Step 2: Incorrect Simplification of Numerators and Denominators

Examine the middle expression \( \frac{2 + 6}{2 + 8} \). This expression means we are adding numbers in the numerator and in the denominator separately. The expression equals \( \frac{8}{10} \), but it's incorrectly simplified as \( \frac{8}{10} \) to \( \frac{6}{8} \). This step didn't consider factoring, but rather a wrong straight addition.

Step 3: Simplification Error

Since \(2\) is cancelled inside the bracket, check if it's correct. Cancelling like this is only valid if all terms can be completely factored by the number outside the bracket. Here that is not valid.

Step 4: Correct Approach

To simplify \(\frac{8}{10}\), divide both the numerator and denominator by their greatest common divisor, which is \(2\). This gives \(\frac{4}{5}\), not \(\frac{3}{4}\) as falsely arrived at when incorrectly manipulating the expression.

Key Concepts

Numerator and Denominator

In a fraction, we have two important components: the numerator and the denominator. These components are positioned in a top/bottom manner. The top number is called the numerator, while the bottom one is termed the denominator. Together, they form the complete fraction, signifying how many parts of a whole we are considering. When working with

• the fraction \( \frac{8}{10} \)
• 8 is the numerator
• 10 is the denominator

It's essential to understand their roles. The numerator indicates the number of equal parts we have. The denominator, on the other hand, provides the total number of equal parts into which the whole is divided. When simplifying fractions, actions are taken to change these values without altering the proportion represented by the fraction.

Equivalent Fractions

Fractions are equivalent when they represent the same portion of a whole, even if the numerators and denominators are different. For instance, the fractions \( \frac{1}{2} \) and \( \frac{2}{4} \) are equivalent because they both indicate the same amount—half of something.To determine if two fractions are equivalent, you can cross-multiply. If the products are equal, the fractions are equivalent. To find equivalent fractions:

• Multiply or divide both the numerator and denominator by the same non-zero number.
• For example, \( \frac{1}{2} \) becomes \( \frac{2}{4} \) when both numbers are multiplied by 2.

Mistakes can occur when we attempt operations that appear to create equivalency but are mathematically unsound—such as adding numbers inside the numerator and denominator separately, as noted in the original solution.

Greatest Common Divisor

The greatest common divisor (GCD), sometimes called the greatest common factor (GCF), is fundamental in simplifying fractions. It's the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD helps us reduce fractions to their simplest form, ensuring they are as straightforward as possible.For instance, to simplify \( \frac{8}{10} \):

• Determine the GCD of 8 and 10, which is 2.
• Divide both the numerator and denominator by this GCD.
• The simplified fraction is \( \frac{4}{5} \).

Recognizing and correctly applying the GCD prevents errors like erroneously simplifying fractions through unsupported shortcuts, as we observed in the original exercise. This concept is crucial for ensuring the accuracy of mathematical operations involving fractions.