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.
You cannot add or subtract fractions with unlike denominators because the denominators represent the total number of equal parts into which a whole is divided. When the denominators of two fractions are different, it means that the wholes are divided into different-sized parts, making them incompatible for direct addition or subtraction.
To understand this better, let's consider a real-world example:
Example 1:
Imagine you have 1 pizza, which you want to share equally among your friends. If the pizza is divided into 8 equal slices, and you give each person 3 slices, you can easily calculate 3/8 of the total pizza shared with each friend. However, if you now have another pizza divided into 6 equal slices and distribute 2 slices to each friend, you cannot directly add or subtract the fractions representing these distributions. The pizzas were divided into different-sized parts, making the fractions incompatible for direct calculation.
On the other hand, when you have fractions with like denominators, it means that the wholes are divided into the same-sized parts. In this case, you can directly add or subtract the fractions by adding or subtracting their numerators while keeping the common denominator.
Example 2:
Suppose in a school cafeteria, there are two trays of cookies. Tray A has 5/8 of its cookies left, and tray B has 3/8 of its cookies left. Since the trays contain the same-sized parts (8 parts in total), you can easily add these fractions together: 5/8 + 3/8 = 8/8. This means that a total of 8 out of the 8 cookie parts are left, which is equivalent to a completely full tray of cookies.
In summary, fractions with unlike denominators cannot be directly added or subtracted as they represent different-sized parts. Fractions with like denominators, however, share the same-sized parts, allowing for direct addition or subtraction operations.