Explain why you cannot add or subtract fractions with unlike denominators while you can add or subtract fractions with like denominators. Include one or more real-world examples in your explanation.(2 points)

Answer 1

Adding or subtracting fractions with like denominators is straightforward because the denominators indicate the same "whole" being divided into equal parts. When the denominators are the same, you simply combine the numerators and keep the denominator unchanged.

For example, if you have two fractions like \( \frac{2}{5} \) and \( \frac{3}{5} \), they both represent parts of the same whole divided into 5 equal parts. You can add them directly:

\[ \frac{2}{5} + \frac{3}{5} = \frac{2 + 3}{5} = \frac{5}{5} = 1 \]

In this case, you're adding 2 parts and 3 parts of the same 5 parts whole, which results in a complete whole.

On the other hand, when fractions have unlike denominators, they represent parts of different wholes, making direct addition or subtraction impossible without first finding a common denominator.

For instance, consider \( \frac{1}{4} \) and \( \frac{1}{3} \). Here, \( \frac{1}{4} \) represents one out of four equal parts of a whole, while \( \frac{1}{3} \) is one out of three equal parts of a different whole. These two fractions cannot simply be combined because they represent different quantities. To find a common value, you would convert both fractions to have a common denominator (which could be 12 in this case):

\[ \frac{1}{4} = \frac{3}{12} \quad \text{and} \quad \frac{1}{3} = \frac{4}{12} \]

Now you can add them:

\[ \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \]

In real-world terms, consider you have a recipe requiring \( \frac{1}{4} \) of a cup of sugar and \( \frac{1}{3} \) of a cup of brown sugar. Since the sugar and brown sugar are measured in different portions, you cannot simply say you have \( \frac{5}{7} \) of a cup without first converting both measurements to a common denominator (in this case, finding a common base for the fractions). This is crucial to combine these quantities meaningfully, so you would convert to a common denominator (like \( \frac{3}{12} \) and \( \frac{4}{12} \)) before adding them.

Thus, adding or subtracting fractions with unlike denominators requires converting to a common denominator to ensure that we are combining equivalent portions of the same whole.