Explain why you cannot add or subtract fractions with unlike denominators while you can add or subtract fractions with like denomjavascript:void(0)inators. Include one or more real-world examples in your explanation

Answer 1

To understand why you can add or subtract fractions with like denominators but not with unlike denominators, let's break it down.

Like Denominators

Fractions with like denominators have the same bottom number (denominator), which represents the same whole. When you add or subtract these fractions, you're essentially combining the same sized pieces.

Example: Imagine you have two pieces of pie. One piece is 1/4 of a pie, and the other is also 1/4 of a pie. If you want to find out how much pie you have in total, you simply add the numerators (the top numbers) together:

1/4 + 1/4 = (1 + 1) / 4 = 2/4 = 1/2

Here, you were able to add the fractions because each piece was the same size—1/4 of the same pie.

Unlike Denominators

Fractions with unlike denominators represent different sized pieces, which means that you can't combine them directly without converting them first.

Example: Suppose you have 1/4 of a pie and 1/3 of another pie. If you want to figure out how much pie you have in total, it isn’t straightforward because these pieces represent different sizes. To add these fractions, you need to find a common denominator (a number that both denominators can be converted into), and rewrite each fraction accordingly.

1. The denominators are 4 and 3. The least common denominator (LCD) is 12.
2. Convert 1/4 into twelfths:
   • 1/4 = 3/12 (because 1 × 3 / 4 × 3 = 3/12)
3. Convert 1/3 into twelfths:
   • 1/3 = 4/12 (because 1 × 4 / 3 × 4 = 4/12)

Now you can add them:

3/12 + 4/12 = (3 + 4) / 12 = 7/12

Real-World Example

Imagine a recipe that requires different fractions of a cup of ingredients. If you have 1/2 cup of sugar and want to add 1/3 cup of brown sugar, you can't just combine them directly because they represent different quantities in terms of capacity.

1. Find a common denominator (which is 6 for 2 and 3).
2. Convert:
   • 1/2 cup of sugar = 3/6
   • 1/3 cup of brown sugar = 2/6
3. Now you can add them together meaningfully:
   • 3/6 + 2/6 = 5/6

In summary, when you have fractions with like denominators, you're adding or subtracting pieces of the same size, which makes calculation straightforward. With unlike denominators, however, the pieces are different sizes, so you need to first find a common denominator to ensure that you're combining like-sized pieces correctly.