In a house made up of 9 rooms, 6 rooms have painted walls, 2 rooms have ceramic tiled walls and 1 room has both painted and ceramic tiled walls. How many rooms do not have either painted or ceramic tiled walls?
(Venn diagram required)
We can use a Venn diagram to represent the rooms with painted walls and the rooms with ceramic tiled walls.
Let P be the set of rooms with painted walls and C be the set of rooms with ceramic tiled walls. The Venn diagram will have two overlapping circles, one for set P and one for set C.
According to the problem, we have the following information:
- |P| = 6 (6 rooms have painted walls)
- |C| = 2 (2 rooms have ceramic tiled walls)
- |P ∩ C| = 1 (1 room has both painted and ceramic tiled walls)
To find the number of rooms without either painted or ceramic tiled walls, we need to find the number of rooms outside of the two circles in the Venn diagram.
Since there is a total of 9 rooms, we can use the principle of inclusion-exclusion to find the number of rooms without either type of walls:
|P ∪ C| = |P| + |C| - |P ∩ C| = 6 + 2 - 1 = 7
Now we can find the number of rooms without either painted or ceramic tiled walls by subtracting the number of rooms with either type of walls from the total number of rooms:
9 - 7 = 2
So, there are 2 rooms that do not have either painted or ceramic tiled walls.
To solve this problem, we can use a Venn diagram to represent the different rooms in the house.
Let's label the circles P and C to represent the painted and ceramic tiled rooms, respectively. The intersection of P and C represents the room that has both painted and ceramic tiled walls. Now, let's fill in the given information:
- P = 6 (rooms with painted walls)
- C = 2 (rooms with ceramic tiled walls)
- P ∩ C = 1 (room with both painted and ceramic tiled walls)
Now, let's calculate the number of rooms that don't have either painted or ceramic tiled walls:
To find the total number of rooms in the house, we need to add the number of rooms in each category and subtract the intersection.
Total rooms = P + C - P ∩ C
Total rooms = 6 + 2 - 1
Total rooms = 7
Therefore, there are 7 rooms in total.
To find the number of rooms that do not have either painted or ceramic tiled walls:
Rooms without either paint or ceramic tiles = Total rooms - (P + C - P ∩ C)
Rooms without either paint or ceramic tiles = 7 - (6 + 2 - 1)
Rooms without either paint or ceramic tiles = 7 - 7 + 1
Rooms without either paint or ceramic tiles = 1
Therefore, there is 1 room that does not have either painted or ceramic tiled walls.
To solve this problem using a Venn diagram, let's first visualize the situation. Draw a rectangle to represent the entire house and divide it into three overlapping circles:
- Circle A represents rooms with painted walls
- Circle B represents rooms with ceramic tiled walls
Now, we know that 6 rooms have painted walls, which means we should place 6 dots inside Circle A. Additionally, we know that 2 rooms have ceramic tiled walls, so we place 2 dots inside Circle B. Lastly, we know that 1 room has both painted and ceramic tiled walls, so we place 1 dot inside the overlapping region of Circle A and Circle B.
After placing the dots, let's count the number of rooms without either painted or ceramic tiled walls. To do this, we need to find the number of empty spaces outside of both circles. In this case, it is the white space outside of Circle A and Circle B.
To find this value, we start with the total number of rooms in the house (9) and subtract the rooms with painted (6), ceramic tiled (2), and the overlapping (1) walls:
9 - 6 - 2 - 1 = 0
Therefore, there are 0 rooms that do not have either painted or ceramic tiled walls in this house.