document.write( "Question 1203656: Thirty gallons of a 20% acid solution is needed for an experiment. Only 40% and 10% acid solutions are available. How much of the 10% acid solution is needed? \n" ); document.write( "

Algebra.Com's Answer #839396 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "First, here is a typical standard formal algebraic solution.

\n" ); document.write( "x gallons of 10% acid, plus (30-x) gallons of 40% acid, makes 30 gallons of 30% acid:

\n" ); document.write( " $\".10%28x%29%2B.40%2830-x%29=.30%2830%29\"$
\n" ); document.write( " $\".1x%2B12-.4x=9\"$
\n" ); document.write( " $\"3=.3x\"$
\n" ); document.write( " $\"x=3%2F.3=10\"$

\n" ); document.write( "ANSWER: x=10 gallons of 10% acid and (30-x)=20 gallons of 40% acid

\n" ); document.write( "And now here are two variations of an informal solution method that is nearly always much faster than formal algebra.

\n" ); document.write( "First variation....

\n" ); document.write( "Picture the three percentages -- 10, 30, and 40 -- on a number line and observe/calculate that 30 is 2/3 of the way from 10 to 40.
\n" ); document.write( "That means 2/3 of the mixture is the higher percentage acid.
\n" ); document.write( "2/3 of 30 gallons is 20 gallons, so again we get the answer of 20 gallons of 40% acid and 10 gallons of 10% acid.
\n" ); document.write( "Second variation....

\n" ); document.write( "Again looking at the three percentages, find that the difference between 10 and 30 is 20 and the difference between 30 and 40 is 10.
\n" ); document.write( "The ratio of those differences, 20 and 10, is 20:10 or 2:1.
\n" ); document.write( "That means the two ingredients must be mixed in the ratio 2:1; and since 30 is closer to 40 than to 10, the large portion must be the 40% acid.
\n" ); document.write( "And 30 gallons in the ratio 2:1 means 20 gallons of the 40% acid and 10 gallons of the 10% acid.

\n" ); document.write( "Here is a crude diagram of how this \"alligation\" method is apparently taught in pharmacology:

\r\n" );
document.write( "\r\n" );
document.write( "    10      10\r\n" );
document.write( "      \    /\r\n" );
document.write( "        30\r\n" );
document.write( "      /    \\r\n" );
document.write( "    40      20

\n" ); document.write( "The two numbers on the left are the percentages of the two ingredients.
\n" ); document.write( "The number in the middle is the percentage of the mixture.
\n" ); document.write( "The two numbers on the right show the ratio (10:20, or 2:1) in which the two ingredients must be mixed, showing that the mixture needs to be 2 parts 40% acid and 1 part 10% acid.
\n" ); document.write( "And 30 gallons with 2 parts 40% acid and 1 part 10% acid means 20 gallons of 40% acid and 10 gallons of 10% acid.

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