Christine, a frequent business traveler, is considering signing up for a hotel rewards program. Hotel Elegance is currently offering 102 points for signing up, plus 78 points for every night Christine books at the hotel. Alternatively, Hotel Paradiso is offering a sign-up bonus of 112 points, plus 77 points per night. If Christine books a certain number of nights, the points she earns with either hotel will be the same. How many points will she have? How many nights will that be?
Let's assume Christine books x nights at both hotels.
For Hotel Elegance:
Points for signing up = 102
Points per night = 78
Total points for Hotel Elegance = 102 + 78x
For Hotel Paradiso:
Points for signing up = 112
Points per night = 77
Total points for Hotel Paradiso = 112 + 77x
Since the total points earned at both hotels will be the same, we can set up the following equation:
102 + 78x = 112 + 77x
Rearranging the equation, we get:
1x = 10
x = 10
Therefore, Christine will have 10 nights at either hotel and will earn:
Total points = 102 + 78x
Total points = 102 + 78(10)
Total points = 102 + 780
Total points = 882
So, she will have 882 points and will stay for 10 nights.