document.write( "Question 994900: A chemist needs 10L of 40% solution of alcohol. He plans to mix a 60% solution with a 20% solution. How many liters of each solution must he mix to produce the required solution? \n" ); document.write( "

Algebra.Com's Answer #613824 by Edwin McCravy(20056) $\"\"$ $\"About$

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A chemist needs 10L of 40% solution of alcohol. He plans to
\n" ); document.write( "mix a 60% solution with a 20% solution. How many liters of
\n" ); document.write( "each solution must he mix to produce the required solution?
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document.write( "Actually in this problem ONLY, since 40% just happens to be \r\n" );
document.write( "exactly half-way between 60% and 20%, we know immediately that \r\n" );
document.write( "it would take exactly the same amount of each.  So we know that\r\n" );
document.write( "to get 10 liters, we must use 5 liters of each.  However, below \r\n" );
document.write( "is the way we would solve such a problem in ALL cases, whether \r\n" );
document.write( "the desired percentage is half-way between the given percentages \r\n" );
document.write( "or not.\r\n" );
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document.write( "Let x = the number of liters of the 40% solution, and\r\n" );
document.write( "Let y = the number of liters of the 60% solution.\r\n" );
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document.write( "Then we make this chart:\r\n" );
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document.write( "                    number        percent       number  \r\n" );
document.write( "                      of          of pure       of liters \r\n" );
document.write( "                    liters        alcohol       of pure\r\n" );
document.write( "                      of          in each       alcohol  \r\n" );
document.write( "                     each         solution      in each  \r\n" );
document.write( "                   solution     as a decimal    solution\r\n" );
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document.write( "First solution        x             0.60        0.60x       \r\n" );
document.write( "Second solution       y             0.20        0.20y\r\n" );
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document.write( "Final solution       10             0.40        0.40(10)\r\n" );
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document.write( "The first equation comes from the number of liters of \r\n" );
document.write( "solution in each solution:\r\n" );
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document.write( "                        x + y = 10\r\n" );
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document.write( "The second equation comes from the the number of liters\r\n" );
document.write( "of pure alcohol in each solution:\r\n" );
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document.write( "                0.60x + 0.20y = .40(10) \r\n" );
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document.write( "We remove the decimals by multiplying each term by 100,\r\n" );
document.write( "which involves moving the decimals two place right.\r\n" );
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document.write( "                    60x + 20y = 40(10)\r\n" );
document.write( "                    60x + 20y = 400\r\n" );
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document.write( "We can simplify that further by dividing through by 20\r\n" );
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document.write( "                       3x + y = 20\r\n" );
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document.write( "Now we have the system of equations\r\n" );
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document.write( "                     \r\n" );
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document.write( "We solve that either by substitution or elimination.  I'll\r\n" );
document.write( "choose substitution.  Solve the first equation for y:\r\n" );
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document.write( "                            y = 10-x\r\n" );
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document.write( "Substitute in the second equation:\r\n" );
document.write( "                      \r\n" );
document.write( "                    3x+(10-x) = 20\r\n" );
document.write( "                      3x+10-x = 20\r\n" );
document.write( "                        2x+10 = 20\r\n" );
document.write( "                           2x = 10\r\n" );
document.write( "                            x = 5 liters of 60%\r\n" );
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document.write( "          Substitute x=5 in y = 10-x\r\n" );
document.write( "                            y = 10-5\r\n" );
document.write( "                            y = 5 liters of 20%\r\n" );
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document.write( "Edwin

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