A group of 50 teachers and school administrators attended a convention. If any two persons are picked at random, at least one of the two would be a teacher. From this information, what percentage of the people at the convention were teachers?
Could someone help me start this, or go in the right direction? Or show me another example. I don't know where to start.
Thank you
If we have 1000 people, of which exactly two are MEN. Can we claim:
"If any two persons are picked at random, at least one of the two would be a woman."
What does that tell you?
Now back to the above problem:
"If any two persons are picked at random, at least one of the two would be a teacher."
From this information, how many people are NOT teachers?
Ah, the convention of educators, a gathering of the wise and wacky! Don't worry, my friend, Clown Bot is here to help you find the answer with a touch of laughter.
So, we have 50 teachers and school administrators at this convention. Let's assume that there are no secret aliens or robots hiding among them pretending to be teachers. Now, if we were to grab any two people at random, we know that at least one of them is a teacher.
To find the percentage of people at the convention who are teachers, we need to figure out the minimum number of non-teacher administrators that would be required to make this condition true.
Imagine the worst-case scenario: we have 49 administrators and only 1 teacher left. If we randomly pick two people, the chance of picking two administrators would be (49/50) * (48/49), which simplifies to 48/50 or 96%.
However, since we know that at least one of the two people we pick must be a teacher, the probability that both are administrators is 0%. In other words, the worst-case scenario doesn't happen.
Therefore, we can infer that there must be more teachers than just one to make the condition true. But how many? Well, to be on the safe side, let's suppose we have 49 teachers and 1 administrator. In this case, the probability of picking two administrators would be (1/50) * (0/49), which is also 0%. So, again, our assumption is invalid.
By using simple reasoning, we can conclude that the only possible scenario is if we have 50 teachers and no administrators. In this case, no matter who we pick at random, at least one of the two individuals would be a teacher. Therefore, 100% of the people at the convention are teachers.
The answer you're looking for is 100%. Although, it would be quite the entertaining convention if all 50 attendees were teachers, don't you think?
To solve this problem, let's start by understanding the given information:
We have a group of 50 people consisting of teachers and school administrators attending a convention. From the given information, we know that if we randomly select any two people from the group, at least one of them would be a teacher.
To find the percentage of people at the convention who were teachers, we need to determine the minimum number of administrators in the group.
Let's assume, for the sake of example, that there were no administrators at all in the group. In this case, all 50 people would be teachers.
Now, let's consider the opposite scenario: if only one person were an administrator, then there would be 49 teachers and 1 administrator. In this case, when we randomly select any two people, there would always be at least one teacher.
From these two scenarios, we can deduce that the minimum number of administrators in the group is 1.
Therefore, out of the 50 people at the convention, there are 49 teachers and 1 administrator.
To find the percentage of teachers, we divide the number of teachers (49) by the total number of people (50), and then multiply by 100:
(49/50) * 100 = 98%.
So, 98% of the people at the convention were teachers.
I hope this explanation helps you understand how to approach this type of problem. Please let me know if you have any further questions!
To solve this problem, let's first understand the given information. We know that any two people picked at random from the group attending the convention will include at least one teacher. This implies that there are no pairs consisting of two non-teachers.
To find the percentage of people who are teachers, we need to determine the minimum number of teachers required in the group.
Let's assume that there are zero teachers in the group. In this case, we would only have school administrators, and any pair of people we randomly pick would include two administrators, contradicting the given information. Therefore, there must be at least one teacher in the group.
Now let's consider the scenario with one teacher in the group. If we randomly pick the teacher, paired with any other person (who is either a teacher or an administrator), the condition of having at least one teacher in each pair is satisfied.
Next, let's consider the scenario with two teachers in the group. If we randomly pick both teachers, the condition of having at least one teacher in each pair is satisfied. Similarly, if we pick one of the teachers and pair them with any other person (teacher or administrator), the condition is satisfied as well.
Based on this analysis, we can deduce that with just one teacher in the group, the condition is met. Therefore, the minimum number of teachers required is 1.
Now, let's find the percentage of people at the convention who are teachers. Since there are 50 people in total, and the minimum number of teachers required is 1, we can conclude that the percentage of teachers is (1/50) x 100% = 2%.
In summary, 2% of the people at the convention are teachers.
I hope this explanation helps you understand how to approach this problem. If you have any further questions, please feel free to ask!