document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204463>Question 1204463</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Pure acid is to be added to a 10​% acid solution to obtain  45L of a 40​% acid solution.<BR>\n" );
document.write( "What amounts of each should be​ used? </SPAN>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "<font color=red>15 liters</font> of pure acid<BR>\n" );
document.write( "<font color=red>30 liters</font> of the 10% acid solution\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "There are at least two ways to solve this problem. <BR>\n" );
document.write( "Method 1 is the standard algebraic approach.<BR>\n" );
document.write( "Method 2 is what is taught to pharmacy students.\r<BR>\n" );
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document.write( "Method 1\r<BR>\n" );
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document.write( "Batch A = pure acid = 100% acid<BR>\n" );
document.write( "Batch B = 10% acid solution\r<BR>\n" );
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document.write( "x = number of liters of batch A<BR>\n" );
document.write( "45-x = number of liters of batch B<BR>\n" );
document.write( "These two amounts combine to 45 liters total.\r<BR>\n" );
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document.write( "x = amount of pure acid from batch A<BR>\n" );
document.write( "0.10(45-x) = 4.5 - 0.10x = amount of pure acid from batch B<BR>\n" );
document.write( "x+(4.5 - 0.10x) = 0.90x+4.5 = total amount of pure acid\r<BR>\n" );
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document.write( "Divide this over the 45 total liters, and the goal is to arrive at a 40% concentration.\r<BR>\n" );
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document.write( "(amount of pure acid)/(total) = concentration<BR>\n" );
document.write( "(0.90x+4.5)/(45) = 0.40<BR>\n" );
document.write( "0.90x+4.5 = 45*0.40<BR>\n" );
document.write( "0.90x+4.5 = 18<BR>\n" );
document.write( "0.90x = 18-4.5<BR>\n" );
document.write( "0.90x = 13.5<BR>\n" );
document.write( "x = 13.5/0.90<BR>\n" );
document.write( "x = <font color=red>15</font><BR>\n" );
document.write( "which leads to<BR>\n" );
document.write( "45-x = 45-15 = <font color=red>30</font>\r<BR>\n" );
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document.write( "Therefore, we need <font color=red>15 liters</font> of pure acid to add to the <font color=red>30 liters</font> of the 10% acid solution.\r<BR>\n" );
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document.write( "In other words, we need <font color=red>15 liters</font> of batch A and <font color=red>30 liters</font> of batch B.\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "Batch A = 100% acid<BR>\n" );
document.write( "Batch B = 10% acid<BR>\n" );
document.write( "target solution = 40%\r<BR>\n" );
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document.write( "Gap from target to 100% is 60%<BR>\n" );
document.write( "Gap from 10% to target is 30%\r<BR>\n" );
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document.write( "The ratio of 60% to 30% is 60:30 = 2:1\r<BR>\n" );
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document.write( "Meaning we'll need twice as much one batch compared to the other.\r<BR>\n" );
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document.write( "2:1 scales up to 2k:1k for some positive real number k.<BR>\n" );
document.write( "2k+1k = 45<BR>\n" );
document.write( "3k = 45<BR>\n" );
document.write( "k = 45/3<BR>\n" );
document.write( "k = 15<BR>\n" );
document.write( "So 2k = 2*15 = 30\r<BR>\n" );
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document.write( "The amounts for batches A and B are 15 and 30. The order isn't clear just yet.\r<BR>\n" );
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document.write( "If we had 30 L of batch A, then we'll have 15*0.10 = 1.5 L of pure acid from batch B.<BR>\n" );
document.write( "The total amount of pure acid would be 30+1.5 = 31.5<BR>\n" );
document.write( "But notice that 31.5/45 = 0.70 = 70% which isn't the 40% target we're after.\r<BR>\n" );
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document.write( "If we had 15 L of batch A, then we'll have 30*0.10 = 3 L of pure acid from batch B.<BR>\n" );
document.write( "15+3 = 18 L of pure acid total out of 45 L total<BR>\n" );
document.write( "18/45 = 0.40 = 40%<BR>\n" );
document.write( "We've reached the goal.\r<BR>\n" );
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document.write( "For more information, search out \"Alligation Method Pharmacy\". <BR>\n" );
document.write( "The spelling of \"Alligation\" is correct. There isn't an \"e\" but an \"i\" instead. Think \"alligator\". <BR>\n" );
document.write( "It's unfortunate the two words sound identical, and are spelled almost the same.\r<BR>\n" );
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document.write( "Here is a useful calculator<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://calculatormaths.com/alligation-calculator\">https://calculatormaths.com/alligation-calculator</a><BR>\n" );
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