document.write( "Question 1208723: Ruth has a beaker containing a solution of $800$ mL of acid and $200$ mL of water. She thinks the solution is a little strong, so she drains $300$ mL from the beaker, adds $300$ mL of water, and stirs the solution. Ruth thinks the solution is still too strong, so again she drains $400$ mL from the beaker, and adds $400$ mL of water. Ruth again thinks the solution is still too strong, so again she drains $100$ mL from the beaker, and adds $100$ mL of water. How many mL of water are now in the beaker? \n" ); document.write( "
Algebra.Com's Answer #847348 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.
\n" ); document.write( "She thinks the solution is a little strong, so she drains 300 mL from the beaker, adds 300 mL of water, and stirs the solution.
\n" ); document.write( "Ruth thinks the solution is still too strong, so again she drains 400 mL from the beaker, and adds 400 mL of water.
\n" ); document.write( "Ruth again thinks the solution is still too strong, so again she drains 100 mL from the beaker, and adds 100 mL of water.
\n" ); document.write( "How many mL of water are now in the beaker?
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\n" ); document.write( "\n" ); document.write( "        The answers in the posts by  Edwin and by  @MathThearapy are different - so,  there is the need to check their solutions.\r
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\n" ); document.write( "\n" ); document.write( "        Here I make this check, but I use slightly different methodology, which better suits to the problem.\r
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\n" ); document.write( "\n" ); document.write( "        In this problem, after each step, Ruth keeps the total volume of the mixture in the beaker unchangeable, 1000 mL.\r
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\n" ); document.write( "\n" ); document.write( "        Therefore, in my solution, I calculate and watch/track for the concentration/amount of the acid, only, in the beaker.\r
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\n" ); document.write( "\n" ); document.write( "        I will calculate the amount of water in the beaker after the last step, but will not calculate/watch/track it at the intermediate steps.\r
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\n" ); document.write( "\n" ); document.write( "        It will allow me to reduce the volume of calculations and will allow me to concentrate\r
\n" ); document.write( "\n" ); document.write( "        my attention on one component, only (which is acid). It will diminish the volume \r
\n" ); document.write( "\n" ); document.write( "        of calculations and will diminish possible errors.\r
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document.write( "(0)  Starting amount of the acid is 800 mL;  the starting concentration is  \"800%2F1000\" = 0.8.\r\n" );
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document.write( "(1)  First step is to drain 300 mL of the mixture from the beaker.\r\n" );
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document.write( "     With it,  0.8*300 = 240 mL of the acid goes out.\r\n" );
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document.write( "     The amount of the acid remained in the beaker is 800-240 = 560 mL.\r\n" );
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document.write( "     The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.\r\n" );
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document.write( "     The concentration after step 1  is  \"560%2F1000\" = 0.56.\r\n" );
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document.write( "(2)  Second step is to drain 400 mL of the mixture from the beaker.\r\n" );
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document.write( "     With it,  0.56*400 = 224 mL of the acid goes out.\r\n" );
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document.write( "     The amount of the acid remained in the beaker is 560-224 = 336 mL.\r\n" );
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document.write( "     The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.\r\n" );
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document.write( "     The concentration after step 2  is  \"336%2F1000\" = 0.336.\r\n" );
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document.write( "(3)  Third step is to drain 100 mL of the mixture from the beaker.\r\n" );
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document.write( "     With it,  0.336*100 = 33.6 mL of the acid goes out.\r\n" );
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document.write( "     The amount of the acid remained in the beaker is 336-33.6 = 302.4 mL.\r\n" );
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document.write( "     The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.\r\n" );
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document.write( "     Hence, the amount of water in the beaker is  1000 - 302.4 = 697.6 mL.\r\n" );
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document.write( "ANSWER.  The amount of water in the beaker after step 3 is  697.6 mL.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "The answer by @MathTherapy is confirmed. \r
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\n" ); document.write( "\n" ); document.write( "Calculations are made by the most economic/effective way.\r
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\n" ); document.write( "\n" ); document.write( "                    Of course,  the most important element of my post \r
\n" ); document.write( "\n" ); document.write( "        is the  METHODOLOGY,  which  EXCLUDES  unnecessary calculations.\r
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\n" ); document.write( "\n" ); document.write( "The answer by Edwin is incorrect.\r
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\n" ); document.write( "\n" ); document.write( "        To help  Edwin to identify his error,  I copied and pasted here his solution\r
\n" ); document.write( "\n" ); document.write( "         from his post,  and pointed the precise place/line with the error.\r
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document.write( "Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.\r\n" );
document.write( "She thinks the solution is a little strong, so she drains 300 mL from the\r\n" );
document.write( "beaker, adds 300 mL of water, and stirs the solution. Ruth thinks the solution\r\n" );
document.write( "is still too strong, so again she drains 400 mL from the beaker, and adds 400 mL\r\n" );
document.write( "of water. Ruth again thinks the solution is still too strong, so again she\r\n" );
document.write( "drains 100 mL from the beaker, and adds 100 mL of water. How many mL of water\r\n" );
document.write( "are now in the beaker?\r\n" );
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document.write( "I'll keep track of the liquid, acid, water, % acid, and % water, because in\r\n" );
document.write( "other problems like this you might be asked different amounts and percentages.\r\n" );
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document.write( "Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water. \r\n" );
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document.write( " liquid = 800+200 = 1000 mL\r\n" );
document.write( "   acid = 800 mL\r\n" );
document.write( "  water = 200 mL\r\n" );
document.write( " % acid = 800/1000 = 0.8 = 80%\r\n" );
document.write( "% water = 200/1000 = 0.2 = 20% \r\n" );
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document.write( "She thinks the solution is a little strong, so she drains 300 mL from the beaker,  \r\n" );
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document.write( " liquid =  1000-300 = 700 mL\r\n" );
document.write( "   acid =  (0.8)(700) = 560 mL\r\n" );
document.write( "  water =  (0.2)(700) = 140 mL\r\n" );
document.write( " % acid =   80%\r\n" );
document.write( "% water =   20% \r\n" );
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document.write( "adds 300 mL of water, and stirs the solution.  \r\n" );
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document.write( " liquid =  700+300 = 1000 mL\r\n" );
document.write( "   acid =  560 mL\r\n" );
document.write( "  water =  140+300 = 440 mL\r\n" );
document.write( " % acid =  560/1000 = 0.56 = 56%\r\n" );
document.write( "% water =  440/1000 = 0.44 = 44% \r\n" );
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document.write( "Ruth thinks the solution is still too strong, so again she drains 400 mL from the beaker,  \r\n" );
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document.write( " liquid =  1000-400 = 600 mL\r\n" );
document.write( "   acid =  (600)(0.56) = 336 mL\r\n" );
document.write( "  water =  (600)(0.44) = 264 mL\r\n" );
document.write( " % acid =  56%\r\n" );
document.write( "% water =  44% \r\n" );
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document.write( "and adds 400 mL of water.  \r\n" );
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document.write( " liquid =  600+400=1000 mL\r\n" );
document.write( "   acid =  336 mL\r\n" );
document.write( "  water =  264+400=664 mL\r\n" );
document.write( " % acid =  336/1000=33.6% \r\n" );
document.write( "% water =  664/1000=66.4% \r\n" );
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document.write( "Ruth again thinks the solution is still too strong, so again she drains 100 mL from the beaker,  \r\n" );
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document.write( " liquid =  1000-100=900 mL\r\n" );
document.write( "   acid =  (900)(0.336) = 302.4 mL   <<<---===  The error is HERE.  The correct calculation should be acid = 1000*0.336 = 336 mL, \r\n" );
document.write( "                                                                    with relevant correction in  all consequential calcs. \r\n" );
document.write( "  water =  664-100=564 mL\r\n" );
document.write( " % acid =  33.6% \r\n" );
document.write( "% water =  66.4% \r\n" );
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document.write( "and adds 100 mL of water.  \r\n" );
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document.write( " liquid =  900+100=1000 mL \r\n" );
document.write( "   acid =  302.4 mL\r\n" );
document.write( "  water =  564+100=664 mL\r\n" );
document.write( " % acid =  302.4/1000 0.3024=30.24% \r\n" );
document.write( "% water =  664/1000 = 0.664 = 66.4% \r\n" );
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document.write( "How many mL of water are now in the beaker? \r\n" );
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document.write( "664 mL   <-- ANSWER\r\n" );
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document.write( "Edwin
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\n" ); document.write( "\n" ); document.write( "This shows once again how dangerous is to split your attention
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\n" ); document.write( "\n" ); document.write( "Professional writers in such cases say: how good that this mistake was made naturally;
\n" ); document.write( "otherwise, it would have been worth to make it intentionally, and then talking, discussing
\n" ); document.write( "and teaching on how to organize calculations in a secure way.\r
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\n" ); document.write( "\n" ); document.write( "    @ikleyn\r
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