All movies until 4: 00 p. m require a discount ticket, and all movies after 4: 00 p. m require a regular-price ticket. each theater at the movieplex seats 98
people. what is the least number of showings the theater could have shown after 4: 00 p. m last saturday ? explain how you found your answer

I'm guessing that there's more information missing from
this question than is given ... quite an accomplishment. 
It's as if I were to ask you:  "I live in Chicago, work as an
engineer, and my car is blue. How old am I ?  Explain how
you found your answer."

We need to know something about how much a discount
ticket costs, and/or how much a regular-price ticket costs,
and/or how much the theater took in at the box office last
Saturday.

Working with all the information given in the question, the
least possible number of showings after 4:00 PM could have
been zero, if the operators shut the doors at 4:00 PM and
went home.


After reading the question again, more carefully this time,
I'll also need to know how many theaters there are in that
movieplex, and if Saturday's box-office take refers to the
whole movieplex, or to only one theater.

127 millimeters is the answer