document.write( "Question 818472: The rectangle has been broken into four smaller rectangles. The area of three
\n" ); document.write( "of the smaller rectangles are shown below. Find the area of the fourth one.
\n" ); document.write( "234
\n" ); document.write( "312
\n" ); document.write( "270
\n" ); document.write( "x \n" ); document.write( "
Algebra.Com's Answer #492636 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
You must have a drawing to go with the problem.
\n" ); document.write( "Otherwise, there could be a lot of answers.
\n" ); document.write( "See NOTE 1 below.
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\n" ); document.write( "If your drawing looks like this:
\n" ); document.write( " and they do not tell you what area belongs to which rectangle.
\n" ); document.write( "You know that one of the small rectangles (rectangle A in my drawing)
\n" ); document.write( "has the same width as a second small rectangle (B), and the same length as a third small rectangle (C).
\n" ); document.write( "If the width and length measures are whole numbers,
\n" ); document.write( "A could measure 18 by 13, with \"area=18%2A13=234\" ,
\n" ); document.write( "B could measure 24 by 13, with \"area=24%2A13=312\" , and
\n" ); document.write( "C could measure 18 by 15, with \"area=18%2A15=270\" .
\n" ); document.write( "In that case, the 4th small rectangle measures 24 by 15,
\n" ); document.write( "with are area of \"24%2A15=highlight%28360%29\" .
\n" ); document.write( "That is one possible solution.
\n" ); document.write( "
\n" ); document.write( "How do we figure that?
\n" ); document.write( "Factoring.
\n" ); document.write( "The possible factorings for the areas are:
\n" ); document.write( "\"234=1%2A236=2%2A117=3%2A78=6%2A39=9%2A26=13%2A18\" ,
\n" ); document.write( "\"312=1%2A312=2=156=3%2A104=4%2A78=6%2A104=8%2A39=12%2A26=13%2A24\" , and
\n" ); document.write( "\"270=1%2A270=2%2A135=3%2A90=5%2A54=6%2A45=9%2A30=10%2A27=15%2A18\" .
\n" ); document.write( "No we look for common factors.
\n" ); document.write( "One of those area values is the area of rectangle A.
\n" ); document.write( "One of the factorings must have
\n" ); document.write( "one factor in common with the factoring for the area of B, and
\n" ); document.write( "the other factor in common with the factoring for the area of C.
\n" ); document.write( "We look for common factors, like \"13\",
\n" ); document.write( "which appears in \"234=13%2A18\" and \"312=13%2A24\" .
\n" ); document.write( "Since we find \"18\" in \"234=13%2A18\" , and
\n" ); document.write( "also as a factor of the third area value, \"270=15%2A18\" ,
\n" ); document.write( "we know that the small rectangles could be as I described above,
\n" ); document.write( "a \"13\" by \"18\" rectangle with area=\"234\" , adjacent to
\n" ); document.write( "a \"13\" by \"24\" rectangle with area=\"312\" , and
\n" ); document.write( "a \"15\" by \"18\" rectangle with area=\"270\" .
\n" ); document.write( "
\n" ); document.write( "However, it could also be that A is
\n" ); document.write( "\"9\" by \"26\" rectangle with area=\"234\" , adjacent to
\n" ); document.write( "a \"9\" by \"30\" rectangle with area=\"270\" , and
\n" ); document.write( "a \"12\" by \"26\" rectangle with area=\"312\" .
\n" ); document.write( "That would make the 4th small rectangle
\n" ); document.write( "a \"12\" by \"30\" rectangle with area=\"360\" .
\n" ); document.write( "Or it could be that A is
\n" ); document.write( "\"6\" by \"39\" rectangle with area=\"234\" , adjacent to
\n" ); document.write( "a \"6\" by \"45\" rectangle with area=\"270\" , and
\n" ); document.write( "a \"39\" by \"45\" rectangle with area=\"312\" .
\n" ); document.write( "That would make the 4th small rectangle
\n" ); document.write( "a \"8\" by \"45\" rectangle with area=\"360\" .
\n" ); document.write( "In all cases we end up with area=\"360\" .
\n" ); document.write( "We could prove why is that with a little algebra,
\n" ); document.write( "but maybe all that was expected is one possible set of dimensions and/or the result.
\n" ); document.write( "
\n" ); document.write( "NOTE 1:
\n" ); document.write( "Let's say your drawing looks like this:
\n" ); document.write( " and they do not tell you what area belongs to which rectangle.
\n" ); document.write( "That is a different problem, and may have more than one solution.
\n" ); document.write( "At least you know that two of the smaller rectangles have the same width,
\n" ); document.write( "and that their lengths add up to the length of the large rectangle.
\n" ); document.write( "
\n" ); document.write( "It could even be a more complicated picture, where none of the rectangles have obvious common dimensions.
\n" ); document.write( "
\n" ); document.write( "I considered those possibilities, but my bet is that your problem is the one I solved above. If that's not the case, ask again.
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\n" ); document.write( "NOTE 2:
\n" ); document.write( "When you post a problem, here or anywhere, try not to leave out important information, like a drawing that comes with the problem.
\n" ); document.write( "Also, when your problem is not answered here because it is too difficult, the child-geniuses in the forum of artofproblemsolving.com can solve it. Just do not expect a very detailed explanation, and do not \"insult\" them by posting a problem that is too easy for the level where you post it. For example, I would post this problem at the middle school level, or at the most basic high school level. \n" ); document.write( "
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