document.write( "Question 766411: A woman is twice as old as her daughter was when she was as old as her daughter is now. When her daughter is as old as she is now, their combined ages will be 180. How old are they now? \n" ); document.write( "
Algebra.Com's Answer #466924 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
A woman is twice as old as her daughter was when she was as old as her daughter is now. When her daughter is as old as she is now, their combined ages will be 180. How old are they now?
\r\n" );
document.write( "Let W = the woman's age now\r\n" );
document.write( "Let D = the daughter's age now\r\n" );
document.write( "Let X = the number of years AGO\r\n" );
document.write( "        talked about in the problem\r\n" );
document.write( "Let Y = the number of years FROM NOW\r\n" );
document.write( "        talked about in the problem.

\n" ); document.write( "a woman is twice as old as her daughter was X YEARS AGO
\r\n" );
document.write( "W = 2(D - X)

\n" ); document.write( "X YEARS AGO WAS when she (THE WOMAN) was as old as her daughter is now.
\r\n" );
document.write( "W - X = D

\n" ); document.write( "when her daughter is as old as she is now WHICH WILL BE Y YEARS FROM NOW
\r\n" );
document.write( "D + Y = W

\n" ); document.write( "Y YEARS FROM NOW, their combined ages will be 180. 
\r\n" );
document.write( "(W + Y) + (D + Y) = 180\r\n" );
document.write( "\r\n" );
document.write( "So we have this system of 4 equations\r\n" );
document.write( "in 4 unknowns: \r\n" );
document.write( "\r\n" );
document.write( "W = 2(D - X) \r\n" );
document.write( "W - X = D\r\n" );
document.write( "D + Y = W\r\n" );
document.write( "(W + Y) + (D + Y) = 180\r\n" );
document.write( "\r\n" );
document.write( "Simplifying each of those equations:\r\n" );
document.write( "\r\n" );
document.write( " W - 2D + 2X      =   0\r\n" );
document.write( " W -  D -  X      =   0\r\n" );
document.write( "-W +  D -    +  Y =   0\r\n" );
document.write( " W +  D      + 2Y = 180\r\n" );
document.write( "\r\n" );
document.write( "how old are they?\r\n" );
document.write( "\r\n" );
document.write( "Solve that by elimination and get\r\n" );
document.write( "\r\n" );
document.write( " W - 2D + 2X      =   0\r\n" );
document.write( " W -  D -  X      =   0\r\n" );
document.write( "-W +  D -    +  Y =   0\r\n" );
document.write( " W +  D      + 2Y = 180\r\n" );
document.write( "\r\n" );
document.write( "Multiply the 2nd equation by 2 and add it to the\r\n" );
document.write( "first equation to eliminate X and get \r\n" );
document.write( "\r\n" );
document.write( " W - 2D + 2X      =   0\r\n" );
document.write( "2W - 2D - 2X      =   0\r\n" );
document.write( "-----------------------\r\n" );
document.write( "3W - 4D           =   0\r\n" );
document.write( "\r\n" );
document.write( "Multiply the 3rd equation by -2 and add the\r\n" );
document.write( "fourth equation to it to eliminate Y and get \r\n" );
document.write( "\r\n" );
document.write( "2W - 2D -    - 2Y =   0\r\n" );
document.write( " W +  D      + 2Y = 180\r\n" );
document.write( "-----------------------\r\n" );
document.write( "3W -  D           = 180\r\n" );
document.write( "\r\n" );
document.write( "So solve the system\r\n" );
document.write( "\r\n" );
document.write( "3W - 4D  =   0\r\n" );
document.write( "3W -  D  = 180\r\n" );
document.write( "\r\n" );
document.write( "by multiplying the first equation by -1\r\n" );
document.write( "\r\n" );
document.write( "-3W + 4D =   0\r\n" );
document.write( " 3W -  D = 180\r\n" );
document.write( "---------------\r\n" );
document.write( "      3D = 180\r\n" );
document.write( "       D = 60\r\n" );
document.write( "\r\n" );
document.write( "Substitute that in\r\n" );
document.write( "\r\n" );
document.write( " 3W -  D = 180 \r\n" );
document.write( " 3W - 60 = 180\r\n" );
document.write( "      3W = 240\r\n" );
document.write( "       W = 80\r\n" );
document.write( "\r\n" );
document.write( "Answer: the Woman is 80 and her daughter is 60.\r\n" );
document.write( "\r\n" );
document.write( "To check we need to find X and Y\r\n" );
document.write( "\r\n" );
document.write( " W - X = D\r\n" );
document.write( "80 - X = 60\r\n" );
document.write( "    -X = -20\r\n" );
document.write( "     X = 20 years ago\r\n" );
document.write( "\r\n" );
document.write( " D + Y = W\r\n" );
document.write( "60 + Y = 80\r\n" );
document.write( "     Y = 20 years from now\r\n" );
document.write( "\r\n" );
document.write( "W = 80, D = 60, X = 20, Y = 20\r\n" );
document.write( "\r\n" );
document.write( "The woman is 80, which is twice as old\r\n" );
document.write( "as her 60-year old daughter was 20 years \r\n" );
document.write( "ago, which is 40.  And 80 is twice 40. That\r\n" );
document.write( "checks.  The woman was was 60 then, which\r\n" );
document.write( "is as old as her daughter is now, which \r\n" );
document.write( "indeed is 60.  So that checks.  20 years \r\n" );
document.write( "from now is when her daughter, who is now \r\n" );
document.write( "60, will be 80 (as old as the woman is now).\r\n" );
document.write( "That checks.  The woman will then be 100, \r\n" );
document.write( "and their combined ages will be 100+80 or \r\n" );
document.write( "180.  Everything checks.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );