document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=184700>Question 184700</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the sum of 4 numbers got by aDDING,SUBTRACTING,DIVIDING AND MULTIPLYING 2 NUMBERS IS 441. FIND THE 2 NUMBERS </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #138632 by </B><A HREF=/tutors/aboutme.mpl?userid=J2R2R><B>J2R2R(94)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=J2R2R><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=J2R2R><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=138632\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the first number be X and the second number be Y.\r<BR>\n" );
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document.write( "We have\r<BR>\n" );
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document.write( "X + Y ..........................(1)\r<BR>\n" );
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document.write( "X – Y ..........................(2)\r<BR>\n" );
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document.write( "X times Y ......................(3)\r<BR>\n" );
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document.write( "X / Y ..........................(4)\r<BR>\n" );
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document.write( "Since the sum is a whole number, then X has to be a multiple of Y (for X/Y to be an integer).\r<BR>\n" );
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document.write( "One of the most obvious choices is if X and Y are equal, so let us try it:\r<BR>\n" );
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document.write( "the first would be 2X\r<BR>\n" );
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document.write( "the second would be 0\r<BR>\n" );
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document.write( "the third would be X^2\r<BR>\n" );
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document.write( "the fourth would be 1\r<BR>\n" );
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document.write( "Solve X^2 + 2X + 1 = 441\r<BR>\n" );
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document.write( "(X + 1)^2 = 441\r<BR>\n" );
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document.write( "X + 1 = +/- 21\r<BR>\n" );
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document.write( "Therefore X = 20 (or -22), and Y is the same as X, which gives:\r<BR>\n" );
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document.write( "X = 20, Y = 20\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "X = -22, Y= -22\r<BR>\n" );
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document.write( "20+20; 20-20; 20x20; 20/20 : 40 + 0 + 400 + 1 = 441\r<BR>\n" );
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document.write( "-22-22; -22--22; -22x-22; -22/-22 : -44 + 0 + 484 + 1 = 441\r<BR>\n" );
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document.write( "-----------------------------------------------------------\r<BR>\n" );
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document.write( "Here is a more in depth solution where X and Y are not assumed equal:\r<BR>\n" );
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document.write( "You can do it without supposing they are equal but remember since X/Y is whole, X is a whole multiple (k) of Y so we could say Let X = kY\r<BR>\n" );
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document.write( "(1)	would be kY + Y\r<BR>\n" );
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document.write( "(2)	would be kY – Y\r<BR>\n" );
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document.write( "(3)	would be kYY = kY^2\r<BR>\n" );
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document.write( "(4)	would be kY/Y = k\r<BR>\n" );
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document.write( "Adding all together would give:\r<BR>\n" );
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document.write( "kY^2 + 2kY + k = 441\r<BR>\n" );
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document.write( "k(Y^2 + 2Y + 1) = 441\r<BR>\n" );
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document.write( "k(Y + 1)^2 = 441\r<BR>\n" );
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document.write( "With 441 and (Y + 1)^2 being squared we need k to also be squared and whole.\r<BR>\n" );
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document.write( "The factors of 441 which are square are 1, 9, 49 and 441\r<BR>\n" );
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document.write( "So by writing the products of 441 as k(Y + 1)^2 using these values for k we have\r<BR>\n" );
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document.write( "1 x 21^2 giving k = 1, Y+1 = +/-21; Y = 20 and X = 20 or Y = -22 and X = -22.\r<BR>\n" );
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document.write( "9 x 7^2 giving k = 9, Y+1 = +/-7; Y = 6 and X = 54 or Y = -8 and X = -72.\r<BR>\n" );
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document.write( "49 x 3^2 giving k = 49, Y+1 = +/-3; Y = 2 and X = 98 or Y = -4 and X = -196.\r<BR>\n" );
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document.write( "441 x 1^2 giving k = 441, Y+1 = +/-1, Y = 0 and X = 0 or Y = -2 and X = -882, but we cannot accept 0 as a value for Y since anything divided by 0 is undefined, so only Y = -2, X = -882 is accepted here.\r<BR>\n" );
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document.write( "Therefore the solutions are:\r<BR>\n" );
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document.write( "X = 	20	;	Y = 	20	;	X + Y =	40	;	X - Y = 	0	;	XY = 	400	;	X/Y = 	1	;	Total = 	441<BR>\n" );
document.write( "X = 	-22	;	Y = 	-22	;	X + Y =	-44	;	X - Y = 	0	;	XY = 	484	;	X/Y = 	1	;	Total = 	441<BR>\n" );
document.write( "X = 	54	;	Y = 	6	;	X + Y =	60	;	X - Y = 	48	;	XY = 	324	;	X/Y = 	9	;	Total = 	441<BR>\n" );
document.write( "X = 	-72	;	Y = 	-8	;	X + Y =	-80	;	X - Y = 	-64	;	XY = 	576	;	X/Y = 	9	;	Total = 	441<BR>\n" );
document.write( "X = 	98	;	Y = 	2	;	X + Y =	100	;	X - Y = 	96	;	XY = 	196	;	X/Y = 	49	;	Total = 	441<BR>\n" );
document.write( "X = 	-196	;	Y = 	-4	;	X + Y =	-200	;	X - Y = 	-192	;	XY = 	784	;	X/Y = 	49	;	Total = 	441<BR>\n" );
document.write( "X = 	-882	;	Y = 	-2	;	X + Y =	-884	;	X - Y = 	-880	;	XY = 	1764	;	X/Y = 	441	;	Total = 	441\r<BR>\n" );
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document.write( "So, although k=1 was an obvious guess where X and Y are equal, it simplifies things but guessing 9, 49 and 441 isn’t so obvious. </SPAN>\n" );
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