There are 25 trays on a table in the cafeteria. Each tray contains a cup only, a plate only, or both a cup and a plate. If 15 of the trays contain cups and 21 of the trays contain plates, how many contain both a cup and a plate?
I assume Venn diagrams/methods are used, but I'm not sure how to handle the intersection of the two in equation form.
21+15 = 36
There are only 25 trays, so 11 must contain both cup and plate.
Now you can draw the Venn diagram to see that the intersection of the two regions must contain 11.
To solve this problem using Venn diagram elements, we need to represent the number of trays containing cups, the number of trays containing plates, and the number of trays containing both a cup and a plate. Let's break it down step by step:
1. Start by drawing two overlapping circles to represent cups and plates. Label one circle as "Cups" and the other as "Plates". The overlapping region will represent trays containing both a cup and a plate.
2. We know that 15 trays contain cups, so write "15" inside the "Cups" circle.
3. We also know that 21 trays contain plates, so write "21" inside the "Plates" circle.
4. Now, we need to find the number of trays that contain both a cup and a plate. This will be represented by the overlapping region.
5. Let's assume that the number of trays containing both a cup and a plate is "x". Write "x" inside the overlapping region.
6. We are given that there are a total of 25 trays, so the sum of the trays in each circle and the overlapping region should equal 25.
7. Write an equation based on the information above to represent the total number of trays: 15 (cups) + 21 (plates) + x (both) = 25.
8. Simplify the equation: 36 + x = 25 => x = 25 - 36 => x = -11.
9. Since the number of trays cannot be negative, our assumption is incorrect. This means that there are no trays that contain both a cup and a plate.
Therefore, none of the trays contain both a cup and a plate.