There are 500 rooms in a multi-floored hotel. However, due to a change in rule, the hotel has to decrease the number of floors by 5 . However, the management is able to put 5 more rooms in each floor. Over all, the number of rooms in the hotel decreases by 10%10%. Find the number of floors and the number of rooms/floor the hotel originally had?⚡a10 floors 50 roomsb20 floors 20 roomsc20 floors 25 roomsd50 floors 10 rooms
Question
There are 500 rooms in a multi-floored hotel. However, due to a change in rule, the hotel has to decrease the number of floors by 5 . However, the management is able to put 5 more rooms in each floor. Over all, the number of rooms in the hotel decreases by 10%10%. Find the number of floors and the number of rooms/floor the hotel originally had?⚡a10 floors 50 roomsb20 floors 20 roomsc20 floors 25 roomsd50 floors 10 rooms
Solution
Let's denote the original number of floors as F and the original number of rooms per floor as R. We know that F * R = 500, which is the total number of rooms in the hotel.
After the change, the number of floors is F - 5 and the number of rooms per floor is R + 5. The total number of rooms after the change is (F - 5) * (R + 5).
We also know that the total number of rooms decreases by 10%, so (F - 5) * (R + 5) = 500 * 0.9 = 450.
Now we have two equations and two unknowns:
- F * R = 500
- (F - 5) * (R + 5) = 450
We can solve these equations by substitution or elimination. If we substitute the first equation into the second, we get:
(F - 5) * (500/F + 5) = 450 500 - 5R + 2500/F - 25 = 450 5R - 2500/F = 50 R - 500/F = 10 R = 10F + 500
Substituting this back into the first equation gives:
F * (10F + 500) = 500 10F^2 + 500F - 500 = 0
Solving this quadratic equation gives F = 20 and R = 25, so the hotel originally had 20 floors and 25 rooms per floor. Therefore, the answer is (c) 20 floors, 25 rooms.
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