document.write( "Question 1090336: Jennifer wants to make 10 gal. of a 40% alcohol solution by mixing together a 10% alcohol solution and a 60% alcohol solution. How much of each solution must she us? This is so confusing please help me out! \n" ); document.write( "

Algebra.Com's Answer #704765 by KMST(5328) $\"\"$ $\"About$

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In math class, \"mixture problems\" are solved as a set of linear equations.
\n" ); document.write( "Defining variables:
\n" ); document.write( " $\"x\"$ = volume of 10% alcohol solution to be used, in gallons
\n" ); document.write( " $\"y\"$ = volume of 60% alcohol solution to be used, in gallons
\n" ); document.write( "Setting up equations:
\n" ); document.write( "Because Jennifer wants to make 10 gallons of solution,
\n" ); document.write( " $\"x%2By=10\"$ .
\n" ); document.write( "Because the final solution must contain 40% alcohol,
\n" ); document.write( "the alcohol amounts in the mixed solutions, must add to the alcohol amount in the final solution.
\n" ); document.write( "The equation stating that is
\n" ); document.write( " $\"0.1x%2B0.6y=0.4%2A10\"$ .
\n" ); document.write( " $\"0.4%2A10\"$ represents the amount of alcohol in 10 gallons of 40% solution,
\n" ); document.write( " $\"0.1x\"$ represents the amount of alcohol in x gallons of 10% solution, and
\n" ); document.write( " $\"0.6y\"$ represents the amount of alcohol in y gallons of 60% solution.
\n" ); document.write( "Those amounts are expressed in the same units, so it does not matter what those units are.
\n" ); document.write( "Solving the system of equations:
\n" ); document.write( " $\"0.1x%2B0.6y=0.4%2A10\"$ would be simplified to $\"0.1x%2B0.6y=4\"$ ,
\n" ); document.write( "and every term would be multiplies times $\"10\"$ to get the equivalent equation
\n" ); document.write( " $\"x%2B6y=40\"$
\n" ); document.write( "Then you would solve the system
\n" ); document.write( " $\"system%28x%2By=10%2Cx%2B6y=40%29\"$
\n" ); document.write( "by whichever way is easier for you.
\n" ); document.write( "Maybe you would choose substitution,
\n" ); document.write( "solving $\"x%2By=10\"$ for $\"x\"$ to find $\"x=10-y\"$ ,
\n" ); document.write( "and then substituting the expression $\"10-y\"$ for $\"x\"$
\n" ); document.write( "in $\"x%2B6y=40\"$ to find $\"y\"$ first.
\n" ); document.write( " $\"10-y%2B6y=40\"$
\n" ); document.write( " $\"10%2B5y=40\"$
\n" ); document.write( " $\"5y=40-10\"$
\n" ); document.write( " $\"5y=30\"$
\n" ); document.write( " $\"y=30%2F5\"$
\n" ); document.write( " $\"highlight%28y=6%29\"$ .
\n" ); document.write( "After that, it would be easiest to substitute the value found
\n" ); document.write( "for $\"y\"$ in equation $\"x=10-y\"$ , to find
\n" ); document.write( " $\"x=10-6\"$
\n" ); document.write( " $\"highlight%28x=4%29\"$ .
\n" ); document.write( "
\n" ); document.write( "NOTE:
\n" ); document.write( "In pharmaceutics class, or in chemical engineering class,
\n" ); document.write( "you would ask what was meant by alcohol.
\n" ); document.write( "Is it ethanol (ethyl alcohol) or isopropyl alcohol?
\n" ); document.write( "You would ask for clarification about the units, and figure out units for the amount of alcohol.
\n" ); document.write( "You would know that when you mix volumes of solutions, the final volume is not exactly the sum of the volumes, but the sum may work as a good approximation for your purposes. \n" ); document.write( "

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