Question

A census-taker asked a hotel owner, "You have reported that you own 5 hotels, and that all the hotels together have a total of 100 rooms. How many rooms are in each hotel?" The owner replied, "Hotels 1 and 2 together contain 52 rooms. Hotels 2 and 3 together contain 43 rooms. Hotels 3 and 4 together contain 34 rooms. And hotels 4 and 5 together contain 30 rooms." "Thanks," said the census-taker. "That gives me all the information I needed." How many rooms are in each hotel? H1=?H2=? H3=? H4=? H5=?

Answer 1

H1 = 9 rooms

H2 = 43 rooms

H3 = 11 rooms

H4 = 23 rooms

H5 = 14 rooms

Step-by-step explanation:

Let's denote the number of rooms in each hotel as H1, H2, H3, H4, and H5. Based on the given information, we can set up a system of equations using the relationships between the total rooms in pairs of hotels.

From the owner's statements:

1. H1 + H2 = 52

2. H2 + H3 = 43

3. H3 + H4 = 34

4. H4 + H5 = 30

Now, we can solve this system of equations. Subtracting equations 1, 2, 3, and 4 successively gives us:

1. H1 = 52 - H2

2. H2 = 43 - H3

3. H3 = 34 - H4

4. H4 = 30 - H5

Substituting these expressions successively into each other yields a value for each hotel:

1. H1 = 52 - (43 - H3) = 9 + H3

2. H2 = 43 - H3

3. H3 = 34 - (30 - H5) = 4 + H5

4. H4 = 30 - H5

Now, combining equations 1, 2, and 3:

- H1 = 9 + (4 + H5) = 13 + H5

Then, considering the total number of rooms in all hotels is 100:

- H1 + H2 + H3 + H4 + H5 = 9 + (43 - H3) + (4 + H5) + (30 - H5) + H5 = 100

- H1 + 43 + 34 + 30 = 100

- H1 = 100 - 43 - 34 - 30 = 100 - 107 = -7

The negative value indicates an error in the equations or initial assumptions. It seems the system might have been set up incorrectly based on the provided information.