document.write( "Question 481764: Will Someone Help Me With This Problem?\r
\n" ); document.write( "\n" ); document.write( "In a survey of four-year colleges and universities, it was found that:\r
\n" ); document.write( "\n" ); document.write( "356 offered a liberal arts degree.
\n" ); document.write( "293 offered a computer engineering degree
\n" ); document.write( "285 offered a nursing degree.
\n" ); document.write( "193 offered a liberal arts degree and a computer engineering degree.
\n" ); document.write( "200 offered a liberal arts degree and a nursing degree.
\n" ); document.write( "139 offered a computer engineering degree and a nursing degree.
\n" ); document.write( "68 offered a liberal arts degree, a computer engineering degree, and a nursing degree.
\n" ); document.write( "26 offered none of these degrees.\r
\n" ); document.write( "\n" ); document.write( "a) How many four-year colleges and universities were surveyed?\r
\n" ); document.write( "\n" ); document.write( "Of the four-year colleges and universities surveyed, how many offered:
\n" ); document.write( "b) a liberal arts degree and a nursing degree, but not a computer engineering degree?
\n" ); document.write( "c) a computer engineering degree, but neither a liberal arts degree nor a nursing degree?
\n" ); document.write( "d) exactly two of these degrees?
\n" ); document.write( "e) at least one of these degrees?\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #329856 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let L = the set of all offering liberal arts, regardless\r\n" );
document.write( "of whether they offered the other two degrees.\r\n" );
document.write( "Let C = the set of all offering computer engineering, regardless\r\n" );
document.write( "of whether they offered the other two degrees.\r\n" );
document.write( "Let N = the set of all offering nursing, regardless\r\n" );
document.write( "of whether they offered the other two degrees.\r\n" );
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document.write( "There are 8 regions which I have labeled with arbitrary small\r\n" );
document.write( "letters.\r\n" );
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document.write( "The most inclusive clue we have is this one:\r\n" );
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\n" ); document.write( "68 offered a liberal arts degree, a computer engineering degree, and a nursing degree
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\r\n" );
document.write( "Those 68 are in all three circles, and the only\r\n" );
document.write( "region that is common to all three circles is\r\n" );
document.write( "the middle region f, so we put 68 for f\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( "193 offered a liberal arts degree and a computer engineering degree
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\r\n" );
document.write( "68 of those 193 are already accounted for in the middle region,\r\n" );
document.write( "so that leaves 193-68 or 125 to go in the other region common\r\n" );
document.write( "to circles L and C, which is region e, so we put 125 in that region:\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( "200 offered a liberal arts degree and a nursing degree.\r
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\r\n" );
document.write( "68 of those 200 are already accounted for in the middle region,\r\n" );
document.write( "so that leaves 200-68 or 132 to go in the other region common\r\n" );
document.write( "to circles L and N, which is region h, so we put 132 in that region:\r\n" );
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document.write( "Now we look at this:\r\n" );
document.write( "

\n" ); document.write( "356 offered a liberal arts degree.
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\r\n" );
document.write( "We have numbers in 3 of the regions of circle L, so \r\n" );
document.write( "132+68+125 or 325 of those 356 are already accounted\r\n" );
document.write( "for so that leaves 356-325 or 31 to go in the remaining\r\n" );
document.write( "region of circle L, region d, so we put 31 in that region:\r\n" );
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document.write( "Now we have completed all four regions of circle L,\r\n" );
document.write( "and see that if we add up all four regions we have\r\n" );
document.write( "31+125+68+132 we have 356 in circle L.\r\n" );
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document.write( "Now we look at this:\r\n" );
document.write( "

\n" ); document.write( "139 offered a computer engineering degree and a nursing degree.
\n" ); document.write( "
\r\n" );
document.write( "68 of those 139 are already accounted for in the middle region,\r\n" );
document.write( "so that leaves 139-68 or 71 to go in the other region common\r\n" );
document.write( "to circles C and N, which is region j, so we put 71 in that region:\r\n" );
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document.write( "\r\n" );
document.write( "Now we look at this:\r\n" );
document.write( "

\n" ); document.write( "293 offered a computer engineering degree
\n" ); document.write( "
\r\n" );
document.write( "We have numbers in 3 of the regions of circle C, so \r\n" );
document.write( "125+68+71 or 264 of those 293 are already accounted\r\n" );
document.write( "for so that leaves 293-264 or 29 to go in the remaining\r\n" );
document.write( "region of circle C, region i, so we put 29 in that region:\r\n" );
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document.write( "Now we look at this:\r\n" );
document.write( "

\n" ); document.write( "285 offered a nursing degree.
\n" ); document.write( "
\r\n" );
document.write( "We have numbers in 3 of the regions of circle N, so \r\n" );
document.write( "132+68+71 or 271 of those 285 are already accounted\r\n" );
document.write( "for so that leaves 285-271 or 14 to go in the remaining\r\n" );
document.write( "region of circle N, region k, so we put 14 in that region:\r\n" );
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document.write( "The most EXcluding clue we have is this one:\r\n" );
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\n" ); document.write( "26 offered none of these degrees
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\r\n" );
document.write( "Those 26 are outside all three circles, and the only\r\n" );
document.write( "remaining region is outside all three circles,\r\n" );
document.write( "the region m, so we put 26 in region m, and our Venn \r\n" );
document.write( "diagram is now complete:\r\n" );
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document.write( "So we are now ready to look at the questions, for now\r\n" );
document.write( "we can answer anything we are asked because we have\r\n" );
document.write( "the complete Venn diagram:\r\n" );
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\n" ); document.write( "a) How many four-year colleges and universities were surveyed?
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document.write( "We add up all the numbers in all 8 regions:\r\n" );
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document.write( "31+125+29+132+68+71+14+26 = 496\r\n" );
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\n" ); document.write( "Of the four-year colleges and universities surveyed, how many offered:
\n" ); document.write( "b) a liberal arts degree and a nursing degree, but not a computer engineering degree?
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\r\n" );
document.write( "That comes from the three regions which are parts of circles L and N,\r\n" );
document.write( "which are not parts of circle C.  Those are the three circles\r\n" );
document.write( "originally labeled d,h, and k, which now contain the numbers\r\n" );
document.write( "31,132, and 14, and when we add 31+132+14 we get 177.  \r\n" );
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\n" ); document.write( "c) a computer engineering degree, but neither a liberal arts degree nor a nursing degree?
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\r\n" );
document.write( "That is the one region that is withing circle C which is not part\r\n" );
document.write( "of either of the other two circles.  That is the upper right region\r\n" );
document.write( "that was originally labeled i, which now contains 29.  \r\n" );
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\n" ); document.write( "d) exactly two of these degrees?
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\r\n" );
document.write( "That would be the three regions originally labeled e,h, and j,\r\n" );
document.write( "which are only parts of two circles, so the answer here is\r\n" );
document.write( "125+132+71 or 328\r\n" );
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\n" ); document.write( "e) at least one of these degrees?
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\r\n" );
document.write( "That is all of the regions which are parts of circles.  The easy way\r\n" );
document.write( "to find it is to take the answer to (a) which was 496 and subtract \r\n" );
document.write( "the 26 that are not in any of the circles and get 496-26 or 470.\r\n" );
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document.write( "Edwin
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