a speaker was so boring that one-half of the audience left after a few minutes. Five minutes later, one-third of the remaing 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
216/7
Well, this is a classic case of a lecture being a snooze-fest! Let's put our clownish math skills to work here. We know that after one-half of the audience left, there were nine people remaining.
So, if we work backward, before one-fourth left, there were nine people. And before one-third left, we just need to add the people who left after one-fourth departed. Well, if one-fourth left and there were nine people left, then four times nine would give us the total number of people remaining at that point.
That would be 36 people. And before one-half left, we just have to add the people who left after one-third departed. So, if one-third left and there were 36 people left, then three times 36 would give us the total number of people remaining earlier.
That would be 108 people. And since that was the beginning of the lecture, there must have been 108 people in the audience when this whole boring affair started. You might want to suggest to the speaker to lighten things up next time!
To solve this problem, we can work backwards by finding the number of people in the audience step by step.
Let's start at the end of the problem where there are only nine people left in the audience. We know that this represents 1/4 of the audience after the third group left.
So, to find the number of people after the third group left, we need to multiply the current number of people (9) by the reciprocal of 1/4, which is 4.
9 * 4 = 36
Therefore, there were 36 people in the audience after the second group left.
Now, let's move to the second step. We know that the remaining 36 people represent 2/3 of the audience after the second group left.
To find the total number of people after the second group left, we divide the current number of people (36) by 2/3. This can be done by multiplying 36 by the reciprocal of 2/3, which is 3/2 or 1.5.
36 * 1.5 = 54
Hence, there were 54 people in the audience after the first group left.
Lastly, let's move to the first step. We know that the remaining 54 people represent 1/2 of the audience after the first group left.
To find the total number of people after the first group left, we divide the current number of people (54) by 1/2. We can do this by multiplying 54 by the reciprocal of 1/2, which is 2.
54 * 2 = 108
Therefore, there were 108 people in the audience at the beginning of the lecture.