document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117924>Question 1117924</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The ratio of freshmen to sophomores to juniors to seniors of a certain high school is 5:5:4:2. Which of the following CANNOT be the total number of the students?\r<BR>\n" );
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document.write( "A. 560<BR>\n" );
document.write( "B. 680<BR>\n" );
document.write( "C. 1,024<BR>\n" );
document.write( "D. 1,600 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733100 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=733100\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your ratio is 5:5:4:2\r<BR>\n" );
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document.write( "if you sum up the ratio, then you get a total of 16.\r<BR>\n" );
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document.write( "that means the ratio of freshman, sophomores, juniors, seniors is equal to 5/16, 5/16, 4/16, 2/16 of the total number of students who attend the school.\r<BR>\n" );
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document.write( "what this says is, that for every 16 students, 5 are freshmen, 5 are sophomores, 4 are juniors, 2 are freshmen.\r<BR>\n" );
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document.write( "the total number of students needs to be an integer since you are dealing with counting the number of students who attend the school.\r<BR>\n" );
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document.write( "take your possible answers and divide them by 16.\r<BR>\n" );
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document.write( "A. 560 / 16 = 35<BR>\n" );
document.write( "B. 680 / 16 = 42.5 ***** not an integer<BR>\n" );
document.write( "C. 1,024 / 16 = 64<BR>\n" );
document.write( "D. 1,600 / 16 = 100\r<BR>\n" );
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document.write( "since the total count of students is an integer in all but selection B, then selection B is the solution where the total number of students is not possible, since the total number of students needs to be an integer.\r<BR>\n" );
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document.write( "when you have a ratio, given in the form that you see, you can convert it into a proportion of the total by adding up the ratios and using that as a divisor.\r<BR>\n" );
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document.write( "your ratio is 5:5:4:2.\r<BR>\n" );
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document.write( "add it up and you get 16.\r<BR>\n" );
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document.write( "the proportion of the total becomes:\r<BR>\n" );
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document.write( "5:16, 5:16, 4:16, 2:16.\r<BR>\n" );
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document.write( "this can also be shown as 5/16, 5/16, 4/16, 2/16 in fractional form.\r<BR>\n" );
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document.write( "note that the proportion of the total and the ratio of each element to the total means the same thing.\r<BR>\n" );
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document.write( "here's a table that shows the results for selection A, C, D.<BR>\n" );
document.write( "you can't do B, because the number of students doesn't add up to be an integer.\r<BR>\n" );
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document.write( "in the table, the number of each type is based on their proportion of the total.\r<BR>\n" );
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document.write( "freshman = 5/16<BR>\n" );
document.write( "junior = 5/16<BR>\n" );
document.write( "sophomore = 4/16<BR>\n" );
document.write( "senior = 2/16\r<BR>\n" );
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document.write( "selection      total   Freshman  sophomore  junior  senior   sum\r\n" );
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document.write( "A              560        175        175      140     70      560\r\n" );
document.write( "B              1024       320        320      256     128     1024\r\n" );
document.write( "C              1600       500        400      400     200     1600\r\n" );
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document.write( "total is what the total number of students is expected to be.<BR>\n" );
document.write( "sum is what they add up to after calculating the proportions of each.<BR>\n" );
document.write( "total = sum as it should.\r<BR>\n" );
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document.write( "another way to look at it is as follows:\r<BR>\n" );
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document.write( "let a = number of freshmen<BR>\n" );
document.write( "let b = number of sophomores<BR>\n" );
document.write( "let c = number of juniors<BR>\n" );
document.write( "let d = number of seniors.\r<BR>\n" );
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document.write( "the ratio of a:b:c:d is equal to 5:5:4:2\r<BR>\n" );
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document.write( "this means a/b = 5/5, a/c = 5/4, a/d = 5/2.\r<BR>\n" );
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document.write( "from a/b = 5/5, you solve for b to get b = a.<BR>\n" );
document.write( "from a/c = 5/4, you solve for c to get c = 4/5 * a<BR>\n" );
document.write( "from a/d = 5/2, you solve for d to get d = 2/5 * a\r<BR>\n" );
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document.write( "since a + b + c + d must be equal to the total number of students, you get:\r<BR>\n" );
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document.write( "a + b + c + d = total of students.\r<BR>\n" );
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document.write( "from the equivalencies you just found, you replace b and c and d with their equivalent values of a to get:\r<BR>\n" );
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document.write( "a + a + 4/5 * a + 2/5 * a = total number of students.\r<BR>\n" );
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document.write( "put everything under a common denominator to get 5/5 * a + 5/5 * a + 4/5 * a + 2/5 * a = total number of students.\r<BR>\n" );
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document.write( "sum them up and you get 16/5 * a = total number of students.\r<BR>\n" );
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document.write( "solve for a to get a = 5/16 * total number of students.\r<BR>\n" );
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document.write( "since you know that b = a, then b = 5/16 * total number of students.\r<BR>\n" );
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document.write( "since you know that c = 4/5 * a, then c = 4/5 * 5/16 * total number of students.<BR>\n" );
document.write( "simplify this to get c = 1/4 * total number of students.<BR>\n" );
document.write( "this is equivalent to 4/16 * total number of students.\r<BR>\n" );
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document.write( "since you know that d = 2/5 * a, then d = 2/5 * 5/16 * total number of students.<BR>\n" );
document.write( "simplify this to get d = 1/8 * total number of students.<BR>\n" );
document.write( "this is equivalent to 2/16 * total number of students.\r<BR>\n" );
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document.write( "there you have it.\r<BR>\n" );
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document.write( "a = 5/16 * total number of students.<BR>\n" );
document.write( "b = 5/16 * total number of students.<BR>\n" );
document.write( "c = 4/16 * total number of students.<BR>\n" );
document.write( "d = 2/16 * total number of students.\r<BR>\n" );
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document.write( "for every 16 students, you have 5 freshmen, 5 sophomores, 4 juniors, 2 seniors.\r<BR>\n" );
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document.write( "the total number of students divided by 16 has to be an integer.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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