After every ninth visit to a restaurant you receive a free beverage . after very twelfth visit you receive a free appetizer . if you visit the restaurant 100 times , on which visits will you receive a free beverage and a free appetizer ? at which visit will you first receive a free beverage and a free appetizer?

What you need to do is find the multiples of 9 and 12 less than 100 and find their common multiples.

in this case, 
multiples of 9 = 18, 27, 36, 45, 54, 63, 72, 81, 90, 99
multiples of 12 = 24, 36, 48, 60, 72, 84, 96

so, the the visit that represents the common multiples are the visit in which you get the both prixes.
 So,
you will recieve free beverages and appetizers on your 36th and 72nd visit.

And, at the 36th visit you will first recieve a free beverages and a appetizer