document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201973>Question 1201973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In survey of 900 workers 600 owned houses,500 owned cars and 345 owned boat,300 owned cars and houses,250 owned houses and boats and 270 owned cars and boats 200 owned the three.find how many workers don't owned any of items, find how many workers owned only two of the items </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836581 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=836581\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Set H = houses<BR>\n" );
document.write( "Set C = cars<BR>\n" );
document.write( "Set B = boats\r<BR>\n" );
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document.write( "Let's draw out a venn diagram<BR>\n" );
document.write( "The uppercase letters represent the circle names/labels<BR>\n" );
document.write( "The lowercase letters are the 8 regions I'll talk about down below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-4%2C4%2C4%2C4%29%2C%0D%0Aline%284%2C4%2C4%2C-4%29%2C%0D%0Aline%284%2C-4%2C-4%2C-4%29%2C%0D%0Aline%28-4%2C-4%2C-4%2C4%29%2C%0D%0A%0D%0Acircle%28-1%2C1%2C2%29%2Ccircle%281%2C1%2C2%29%2Ccircle%280%2C-0.7321%2C2%29%2C%0D%0A%0D%0Alocate%28-3.2%2C2.8%2CH%29%2C%0D%0Alocate%282.6%2C2.8%2CC%29%2C%0D%0Alocate%281%2C-2.6%2CB%29%2C%0D%0Alocate%28-2%2C1.8%2Ca%29%2C%0D%0Alocate%280%2C1.8%2Cb%29%2C%0D%0Alocate%282%2C1.8%2Cc%29%2C%0D%0Alocate%28-1.5%2C0%2Cd%29%2C%0D%0Alocate%280%2C0.4%2Ce%29%2C%0D%0Alocate%281.2%2C0%2Cf%29%2C%0D%0Alocate%280%2C-1.5%2Cg%29%2C%0D%0Alocate%283%2C-1.5%2Ch%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are 8 distinct regions<ul><li>a = houses only</li><li>b = houses and cars, but no boats</li><li>c = cars only</li><li>d = houses and boats, but no cars</li><li>e = all three</li><li>f = cars and boats, but no houses</li><li>g = boats only</li><li>h = none of the three items mentioned</li></ul>Example: If someone owns a house and a boat, then they are in region d.\r<BR>\n" );
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document.write( "We have these given facts<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Number</TD><TD >Statement</TD></tr><tr><TD ><font color=blue> Fact 1</font></TD><TD >Survey of 900 workers</TD></tr><tr><TD ><font color=blue> Fact 2</font></TD><TD >600 owned houses</TD></tr><tr><TD ><font color=blue> Fact 3</font></TD><TD >500 owned cars</TD></tr><tr><TD ><font color=blue> Fact 4</font></TD><TD >345 owned boat</TD></tr><tr><TD ><font color=blue> Fact 5</font></TD><TD >300 owned cars and houses</TD></tr><tr><TD ><font color=blue> Fact 6</font></TD><TD >250 owned houses and boats</TD></tr><tr><TD ><font color=blue> Fact 7</font></TD><TD >270 owned cars and boats</TD></tr><tr><TD ><font color=blue> Fact 8</font></TD><TD >200 owned the three.</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<font color=blue>Fact 8</font> is where we'll start. <BR>\n" );
document.write( "This tells us that 200 goes in region e at the very center. <BR>\n" );
document.write( "It's where all three circles overlap.\r<BR>\n" );
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document.write( "We'll then use <font color=blue>Fact 7</font> and <font color=blue>Fact 8</font> to determine there must be 270-200 = 70 people who own a car and a boat, but not a house. <BR>\n" );
document.write( "The value 70 goes inside region f.\r<BR>\n" );
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document.write( "Next use <font color=blue>Fact 6</font> and <font color=blue>Fact 8</font> together to determine there are 250-200 = 50 people who own a house and a boat, but no car.<BR>\n" );
document.write( "The value 50 goes in region d.\r<BR>\n" );
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document.write( "Next up will be <font color=blue>Fact 5</font> and <font color=blue>Fact 8</font> used together. We can say there are 300-200 = 100 people who own a house and a car, but no boat.<BR>\n" );
document.write( "The value 100 goes in region b.\r<BR>\n" );
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document.write( "Here's what we have so far<BR>\n" );
document.write( "b = 100<BR>\n" );
document.write( "d = 50<BR>\n" );
document.write( "e = 200<BR>\n" );
document.write( "f = 70<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-4%2C4%2C4%2C4%29%2C%0D%0Aline%284%2C4%2C4%2C-4%29%2C%0D%0Aline%284%2C-4%2C-4%2C-4%29%2C%0D%0Aline%28-4%2C-4%2C-4%2C4%29%2C%0D%0A%0D%0Acircle%28-1%2C1%2C2%29%2Ccircle%281%2C1%2C2%29%2Ccircle%280%2C-0.7321%2C2%29%2C%0D%0A%0D%0Alocate%28-3.2%2C2.8%2CH%29%2C%0D%0Alocate%282.6%2C2.8%2CC%29%2C%0D%0Alocate%281%2C-2.6%2CB%29%2C%0D%0Alocate%28-2%2C1.8%2Ca%29%2C%0D%0Alocate%280%2C1.8%2C100%29%2C%0D%0Alocate%282%2C1.8%2Cc%29%2C%0D%0Alocate%28-1.5%2C0%2C50%29%2C%0D%0Alocate%280%2C0.4%2C200%29%2C%0D%0Alocate%281.2%2C0%2C70%29%2C%0D%0Alocate%280%2C-1.5%2Cg%29%2C%0D%0Alocate%283%2C-1.5%2Ch%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Focus your attention on circle H.<BR>\n" );
document.write( "The values in this circle must add to 600 (due to <font color=blue>Fact 2</font>)<BR>\n" );
document.write( "That means...<BR>\n" );
document.write( "a+b+d+e = 600<BR>\n" );
document.write( "a+100+50+200 = 600<BR>\n" );
document.write( "a+350 = 600<BR>\n" );
document.write( "a = 600-350<BR>\n" );
document.write( "a = 250<BR>\n" );
document.write( "There are 250 people who own a house, but not a car and also not a boat.\r<BR>\n" );
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document.write( "Now move your attention to circle C.<BR>\n" );
document.write( "We use <font color=blue>Fact 3</font> to say the following:<BR>\n" );
document.write( "b+c+e+f = 500<BR>\n" );
document.write( "100+c+200+70 = 500<BR>\n" );
document.write( "c+370 = 500<BR>\n" );
document.write( "c = 500-370<BR>\n" );
document.write( "c = 130\r<BR>\n" );
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document.write( "Use a similar idea for circle B. Refer to <font color=blue>Fact 4</font><BR>\n" );
document.write( "d+e+f+g = 345<BR>\n" );
document.write( "50+200+70+g = 345<BR>\n" );
document.write( "320+g = 345<BR>\n" );
document.write( "g = 345-320<BR>\n" );
document.write( "g = 25\r<BR>\n" );
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document.write( "We have these values<BR>\n" );
document.write( "a = 250, b = 100, c = 130<BR>\n" );
document.write( "d = 50, e = 200, f = 70<BR>\n" );
document.write( "g = 25<BR>\n" );
document.write( "which leads to this nearly completed venn diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-4%2C4%2C4%2C4%29%2C%0D%0Aline%284%2C4%2C4%2C-4%29%2C%0D%0Aline%284%2C-4%2C-4%2C-4%29%2C%0D%0Aline%28-4%2C-4%2C-4%2C4%29%2C%0D%0A%0D%0Acircle%28-1%2C1%2C2%29%2Ccircle%281%2C1%2C2%29%2Ccircle%280%2C-0.7321%2C2%29%2C%0D%0A%0D%0Alocate%28-3.2%2C2.8%2CH%29%2C%0D%0Alocate%282.6%2C2.8%2CC%29%2C%0D%0Alocate%281%2C-2.6%2CB%29%2C%0D%0Alocate%28-2%2C1.8%2C250%29%2C%0D%0Alocate%280%2C1.8%2C100%29%2C%0D%0Alocate%282%2C1.8%2C130%29%2C%0D%0Alocate%28-1.5%2C0%2C50%29%2C%0D%0Alocate%280%2C0.4%2C200%29%2C%0D%0Alocate%281.2%2C0%2C70%29%2C%0D%0Alocate%280%2C-1.5%2C25%29%2C%0D%0Alocate%283%2C-1.5%2Ch%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The last thing to figure out is what we put in region h.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Let's add up all of the values in the circles ('a' through 'g')<BR>\n" );
document.write( "a+b+c+d+e+f+g = 250+100+130+50+200+70+25 = 825\r<BR>\n" );
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document.write( "This represents the total number of people who have at least one (i.e. one or more) item mentioned.\r<BR>\n" );
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document.write( "Then we use <font color=blue>Fact 1</font> to determine how many people have none of those three items<BR>\n" );
document.write( "900-825 = 75<BR>\n" );
document.write( "There are 75 people who own neither a house, nor a car, nor a boat.<BR>\n" );
document.write( "This value goes in region h.\r<BR>\n" );
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document.write( "Here's the completed venn diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-4%2C4%2C4%2C4%29%2C%0D%0Aline%284%2C4%2C4%2C-4%29%2C%0D%0Aline%284%2C-4%2C-4%2C-4%29%2C%0D%0Aline%28-4%2C-4%2C-4%2C4%29%2C%0D%0A%0D%0Acircle%28-1%2C1%2C2%29%2Ccircle%281%2C1%2C2%29%2Ccircle%280%2C-0.7321%2C2%29%2C%0D%0A%0D%0Alocate%28-3.2%2C2.8%2CH%29%2C%0D%0Alocate%282.6%2C2.8%2CC%29%2C%0D%0Alocate%281%2C-2.6%2CB%29%2C%0D%0Alocate%28-2%2C1.8%2C250%29%2C%0D%0Alocate%280%2C1.8%2C100%29%2C%0D%0Alocate%282%2C1.8%2C130%29%2C%0D%0Alocate%28-1.5%2C0%2C50%29%2C%0D%0Alocate%280%2C0.4%2C200%29%2C%0D%0Alocate%281.2%2C0%2C70%29%2C%0D%0Alocate%280%2C-1.5%2C25%29%2C%0D%0Alocate%283%2C-1.5%2C75%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where,<BR>\n" );
document.write( "a = 250, b = 100, c = 130<BR>\n" );
document.write( "d = 50, e = 200, f = 70<BR>\n" );
document.write( "g = 25, h = 75\r<BR>\n" );
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document.write( "Now onto the questions:\r<BR>\n" );
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document.write( "Question 1) <font color=blue>how many workers don't own any of those items?</font><BR>\n" );
document.write( "Answer: <font color=red size=4>75</font><BR>\n" );
document.write( "Reason: See region h of the venn diagram\r<BR>\n" );
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document.write( "Question 2) <font color=blue>how many workers own only two of the items?</font><BR>\n" );
document.write( "Answer: <font color=red size=4>220</font><BR>\n" );
document.write( "Reason: b+d+f = 100+50+70 = 220<BR>\n" );
document.write( "Ignore region e because that subset of people own all 3 items.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );