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document.write( "There are three ways to do the problem: Venn diagram method, \r\n" );
document.write( "Chart method, and formula method:\r\n" );
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document.write( "Venn diagram method:\r\n" );
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document.write( "Draw a big rectangle to say, represent a big football field.\r\n" );
document.write( "Assume we start with all 350 students seated in the stands. \r\n" );
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document.write( "Now draw a big red circle on the football field for all the\r\n" );
document.write( "biology students to come stand in. Label the circle B for\r\n" );
document.write( "\"biology\"\r\n" );
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document.write( "Next draw a big green circle overlapping the red one for all the\r\n" );
document.write( "chemistry students to come stand in. Label the green circle C for\r\n" );
document.write( "\"chemistry\"\r\n" );
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document.write( "Next notice that there are 4 regions for the students to come down\r\n" );
document.write( "and stand in.\r\n" );
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document.write( "I will begin by labeling them I, II, III and Iv\r\n" );
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document.write( "Now we are going to ask all the students to come down from the stands and\r\n" );
document.write( "stand in which of the four regions they belong in depending of what\r\n" );
document.write( "course or courses they study or don't study: \r\n" );
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document.write( "Let's identify each of those four regions. \r\n" );
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document.write( "Region I:\r\n" );
document.write( "The students who will stand in region I are those who study biology but\r\n" );
document.write( "who do not study chemistry. Notice that they will be inside the red circle\r\n" );
document.write( "but outside the green circle.\r\n" );
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document.write( "Region II:\r\n" );
document.write( "The students who will stand in region II are those who study both biology and\r\n" );
document.write( "chemistry. Notice that they will be inside the red circle and also\r\n" );
document.write( "inside the green circle.\r\n" );
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document.write( "Region III:\r\n" );
document.write( "The students who will stand in region III are those who study chemistry but\r\n" );
document.write( "who do not study biology. Notice that they will be inside the green circle\r\n" );
document.write( "but outside the red circle.\r\n" );
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document.write( "Region IV:\r\n" );
document.write( "The students who will stand in region IV are those who study neither biology\r\n" );
document.write( "nor chemistry. Notice that they will be outside both circles\r\n" );
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document.write( "The object of the problem is to determine how many are in region IV.\r\n" );
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document.write( "We begin by getting the students who are taking both subjects to stand\r\n" );
document.write( "in region II. We are given how many will stand in region II by these \r\n" );
document.write( "words:\r\n" );
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document.write( ">>...78 students are studying both subjects...<<\r\n" );
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document.write( "Those 78 students will all come down and stand in region II since they must\r\n" );
document.write( "stand in both circles. So we'll write 78 where the II is to represent the\r\n" );
document.write( "78 students that are in both circles because they take both courses:\r\n" );
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document.write( "Next we will fill in region I, which is the rest of the red circle.\r\n" );
document.write( "These words:\r\n" );
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document.write( ">>...213 students are enrolled in biology...<<\r\n" );
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document.write( "tell us that 213 students must stand in the red circle. We have already\r\n" );
document.write( "accounted for 78 of them, so that leaves 213-78 or 135 students to stand in\r\n" );
document.write( "region I. So we'll write 135 where the I is to represent the\r\n" );
document.write( "135 students that are in the red circles that are not in the green circle.\r\n" );
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document.write( "
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document.write( "Next we will fill in region III, which is the rest of the green circle.\r\n" );
document.write( "These words:\r\n" );
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document.write( ">>...155 students are studying chemistry...<<\r\n" );
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document.write( "tell us that 155 students must stand in the green circle. We have already\r\n" );
document.write( "accounted for 78 of them, so that leaves 155-78 or 77 students to stand in\r\n" );
document.write( "region III. So we'll write 77 where the III is to represent the\r\n" );
document.write( "78 students that are in the green circle that are not in the red circle.\r\n" );
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\r\n" );
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document.write( "Now finally we'll tell all the rest of the students to stand in region IV.\r\n" );
document.write( "That's the region where all the students who do not take either biology\r\n" );
document.write( "or chemistry will stand.\r\n" );
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document.write( "We know that there are 350 students altogether. We have accounted for \r\n" );
document.write( "135+78+77 or 290 students who take biology or chemistry. So to find\r\n" );
document.write( "out how many will stand outside both circles in region IV, we subtract\r\n" );
document.write( "350 - 290 and get 60. So the remaining 60 will stand in region IV, outside\r\n" );
document.write( "both circles.\r\n" );
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document.write( "So the answer is 60.\r\n" );
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document.write( "Chart method (self-explanator if you understand the Venn diagram:\r\n" );
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document.write( " Taking Chemistry\r\n" );
document.write( " Yes No Totals\r\n" );
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document.write( " | | || \r\n" );
document.write( "Taking YES | 78 | 135 || 213\r\n" );
document.write( "Biology ------|----------|------------||--------- \r\n" );
document.write( " NO | 77 | 60 || 137\r\n" );
document.write( " | | ||\r\n" );
document.write( " ------|-----------------------||---------\r\n" );
document.write( " ------------------------------||--------- \r\n" );
document.write( "Totals | 155 | 195 || 350\r\n" );
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document.write( "Formula method\r\n" );
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document.write( "N(B or C) = N(B) + N(C) - N(B and C)\r\n" );
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document.write( "N(B or C) = 213 + 155 - 78 \r\n" );
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document.write( "N(B or C) = 290\r\n" );
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document.write( "N(Neither B nor C) = N(Universal set) - N(B or C)\r\n" );
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document.write( "N(Neither B nor C) = 350 - 290\r\n" );
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document.write( "N(Neither B nor C) = 60\r\n" );
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document.write( "Edwin
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
