How Many are in the Group?Date: 10/17/96 at 14:58:25 From: Paul Benedict Subject: How Many in Group? Mr. Jackson polled the class to see how many students had been to one or both of the nearby state parks. He found that everyone in Katie's group had been to at least one of the parks. When he asked how many had been to Punta de las Cuevas, four children raised their hands. When he asked how many had been to Salt Bay, three raised their hands. When he asked how many had been to both state parks, two raised their hands. How many people were in Katie's group? Date: 10/18/96 at 17:53:7 From: Doctor Dennis Subject: Re: How Many in Group? Questions like this are often very easy using Venn Diagrams. Imagine you have a big rectangle. In that big rectangle are two circles which are partially overlapping. Now, say that all the students at the table are represented by the stuff within the rectangle. Call one circle people who have gone to Punta de Cuevas and the other circle people who have gone to Salt Bay. Now, since two have been to both parks there are two students in the area that is the intersection of the circles. The Salt bay circle should have 3 people in it, but there are already two people in the part of the circle that intersects with the other circle, so there is just one person in the part of the Salt Bay Circle that is not within the other circle. Similarly, there will be 2 people in the punta de cuevas circle that are not in the part that overlaps the Salt Bay circle. so there are 2 + 2+ 1 = 5 kids total. -Doctor Dennis, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 10/18/96 at 17:52:11 From: Doctor Keith Subject: Re: How Many in Group? Hi Paul, This is an example of everyone's favorite type of problem, the word problem. The real trick to word problems is trying to turn the words into equations. Let's summarize what we have in a table for easy reference: Number of students Place who have been there Punta de las Cuevas 4 Salt Bay 3 Both 2 Neither 0 We have several ways of looking at this but probably the easiest is a logical argument. The two students who have been to both parks would have raised their hands for both Punta de las Cuevas and Salt Bay. Thus two of the students in the Punta de las Cuevas group are also in the Salt Bay group. So let's find out how many have been to only one. For Punta de las Cuevas, four students have been but two also went to Salty Bay, so we have to remove them (remember we want students who have only been to Punta de las Cuevas). 4 - 2 = 2 no. of students who have only been to Punta de las Cuevas Thus only two students have only been to Punta de las Cuevas. Similarly, for Salt Bay we have three students who have gone, but two also went to Punta de las Cuevas so we must remove those two from the three who have been. 3 - 2 = 1 no. of students who have only been to Salt Bay Thus only one student has gone to only Salt Bay. To get the final answer we note that the sum of the students who have only been to Punta de las Cuevas, plus those who have only been to Salty Bay, plus those who have been to both must be the total in the group. So: 2 + 1 + 2 = 5 Thus there are 5 students in the group. If this seemed hard to follow you can always name the kids in small groups like this and count the distinct names. This is a good visualization tool, but I suggest you not get used to it, because it could be a real problem if the numbers got large. To help you see this problem, though, let's run through it. We will revise our table by naming students in the first group only. Place Students who have been there Punta de las Cuevas Katie, Billy, Tom, Sarah Salt Bay 3 Both 2 Neither 0 Now we know that two of the kids from the first group must be in the "Both" group. Let's pick the first two for simplicity. Place Students who have been there Punta de las Cuevas Katie, Billy, Tom, Sarah Salt Bay 3 Both Katie, Billy Neither 0 We also know that the two from the "Both" group must also be in the Salt Bay group, with one student left over to be named as we choose. Place Students who have been there Punta de las Cuevas Katie, Billy, Tom, Sarah Salt Bay Katie, Billy, Fred Both Katie, Billy Neither 0 Thus since we don't want to count Katie and Billy in all three groups, we remove them form the first two and get Place Students who have been there only Punta de las Cuevas Tom, Sarah only Salt Bay Fred Both Katie, Billy Neither 0 And by counting the names we get 5 students. As you can see, while this will help you to see what we did and why, if we were to follow this for a whole class or school, it could easily become a nightmare. Hope this helps you out. Good luck! -Doctor Keith, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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