Final answer:
Adding fractions requires a common denominator because it ensures that the parts being combined are the same size, analogous to adding pieces of a pie cut into the same number of slices. Without a common denominator, we cannot accurately determine how much of a whole we have.
Step-by-step explanation:
When dealing with fractions, it's crucial to understand that the addition of fractions requires a common denominator because you are combining parts of a whole that are the same size. Imagine you have pieces of two different pies, one cut into fourths and another into eighths; combining a quarter of one pie with an eighth of another doesn't give you a clear picture of how much pie you have without converting them into equivalent sections, perhaps eighths, where two eighths (from the quarter) plus one eighth equals three eighths in total.
To add fractions, like 1/2 and 1/3, a common denominator must be found (½ becomes ⅔ and ⅓ becomes ⅖ when the common denominator is 6). We add the adapted numerators (2 + 2 = 4), while the denominators remain unchanged, giving us ⅔ as the final answer. Common denominators enable direct comparison and accurate combination of different fractions. Without them, we would get nonsensical answers like 2/5 (½ + 1/3 without common denominators), which is incorrect.
In the case of multiplying fractions, we simply multiply the numerators together and the denominators together, simplifying as needed. For example, multiplying ⅒ by ⅗ gives us a numerator of 30 and a denominator of 120. After simplification by common factors, the final answer is 1/4. This highlights the importance of simplification, which contributes to clarity and accuracy in our mathematical operations.