A ticket agency sold 1/2 of a concert's tickets in the first hour of sale. After 30 more tickets were sold by the second hour, 1/4 of the original number of tickets remained. How many tickets were there to begin with?

so initially if there were X number of tickets by the first hour only x/2 tickets were left
by the second hour x/2-30 tickets were left which is equal to 1/4 of x

so there are 2 ways
1. logically- we know tickets were half of x after first hour and when 30 more were sold they became 1/2 of x/2 so 30=1/2 of x/2 therefore x/2=60 and so x=120
2. mathematically- using the equation x/2-30=x/4 which when solved for X would give same answer x=120