A speaker was so boring that one-half of the audience left after a few minutes. Five minutes later, one-third of the remaining audience left. Ten minutes later, one-fourth of those remaining left, leaving only nine people in the audience. How many people were in the audience at the beginning of the lecture.

x = people in audience

1/2 x = left 1st
1/3 (1/2 x) = 1/6 x = left next
1/4 (1/3)(1/2)x = 1/24 x = left last

x - (1/2 x + 1/6 x + 1/24 x) = 9

Solve for x

To solve this problem, let's work backwards and use a step-by-step approach.

Step 1: At the end of the lecture, there were only 9 people remaining in the audience.
Step 2: Ten minutes before this, one-fourth of the remaining audience left. So, if we add the 9 people to this quarter, we will get the number of people remaining at that time.
Number of people remaining = 9 + (9 / 1/4) = 9 + (9 * 4) = 9 + 36 = 45
Step 3: Five minutes before this, one-third of the remaining audience left. Following the same logic as above, we can calculate the number of people remaining at that time:
Number of people remaining = 45 + (45 / 1/3) = 45 + (45 * 3) = 45 + 135 = 180
Step 4: At the beginning of the lecture, one-half of the audience left. Again, using the same logic, we can calculate the total number of people in the audience initially:
Total number of people = 180 + (180 / 1/2) = 180 + (180 * 2) = 180 + 360 = 540

Therefore, at the beginning of the lecture, there were 540 people in the audience.