document.write( "Question 652559: A 20% solution of fertilizer is to be mixed with a 80% solution of fertilizer in order to get 120 gallons of a 60% solution. How many gallons of the 20% solution and 80% solution should be mixed? \n" ); document.write( "

Algebra.Com's Answer #408040 by KMST(5328) $\"\"$ $\"About$

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Let the number of gallons of 20% solution of fertilizer to be used be $\"x\"$ .
\n" ); document.write( "Let the number of gallons of 80% solution of fertilizer to be used be $\"y\"$ .\r
\n" ); document.write( "\n" ); document.write( "As you mix the solutions, the volumes add up,
\n" ); document.write( "so the final volume (in gallons) is $\"x%2By=120\"$ .
\n" ); document.write( "That is your first equation.
\n" ); document.write( " $\"x\"$ gallons of a 20% solution contain $\"0.20x\"$ gallons of fertilizer.
\n" ); document.write( " $\"y\"$ gallons of a 80% solution contain $\"0.80y\"$ gallons of fertilizer.
\n" ); document.write( "When you mix them, the final mix will contain $\"0.20x%2B0.80y\"$ gallons of fertilizer,
\n" ); document.write( "but that must be the amount of fertilizer in 120 gallons of 60% solution of fertilizer,
\n" ); document.write( "which is $\"0.60%2A120=72\"$ gallons of fertilizer.
\n" ); document.write( "So $\"0.20x%2B0.80y=72\"$ or $\"0.2x%2B0.8y=72\"$ (which means the same in math class).
\n" ); document.write( "That is your second equation.
\n" ); document.write( "You now have a system of 2 equations linear in two variables:
\n" ); document.write( " $\"system%28x%2By=120%2C0.2x%2B0.8y=72%29\"$
\n" ); document.write( "You can solve it by whatever method you chose.
\n" ); document.write( "For example, to solve by elimination, you may chose to multiply the first equation by $\"%28-0.2%29\"$ , getting
\n" ); document.write( " $\"-0.20x-0.20y=24\"$ ,
\n" ); document.write( "and add it to the second equation, getting
\n" ); document.write( " $\"0.2x%2B0.8y-0.20x-0.20y=72-24\"$ --> $\"0.6x=48\"$ --> $\"x=48%2F0.6%29%29%29+--%3E+%7B%7B%7Bhighlight%28x=80%29\"$
\n" ); document.write( "Now, from $\"x=80\"$ and $\"x%2By=120\"$ , you get
\n" ); document.write( " $\"80%2By=120\"$ --> $\"y=120-80\"$ --> $\"highlight%28y=40%29\"$
\n" ); document.write( "
\n" ); document.write( "NOTES FROM A CHEMIST
\n" ); document.write( "In real life, volume is often additive, so if you mix $\"x\"$ gallons of a solution with $\"y\"$ gallons of another solution, you get $\"x%2By\"$ gallons of mixture. (In math problems, volume is always additive).Chemists (like me) specify very clearly the concentration of a solution.
\n" ); document.write( "If we say 20% weight in weight (w/w), we mean that 100 g of solution contains 20 g of the item of interest.
\n" ); document.write( "If we say 20% weight in volume (w/v), we mean that 100 milliliters of solution contains 20 g of the item of interest.
\n" ); document.write( "If we say 20% volume in volume (v/v), we mean that 100 gallons of solution contains 20 gallons of a liquid of interest.
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