document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175368>Question 1175368</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A chemist has three different acid solutions. The first acid solution contains 15% acid, the second contains 30% and the third contains 50%. They want to use all three solutions to obtain a mixture of 96 liters containing  35% acid, using 3 times as much of the 50% solution as the 30% solution. How many liters of each solution should be used? </SPAN>\n" );
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document.write( "On the chance that you will understand it, I will solve this problem quickly and easily with logical reasoning and simple mental arithmetic -- without the need to set up and solve an algebraic equation.<br><BR>\n" );
document.write( "If you understand what I am doing in this solution, you will have a quick and easy way to solve a wide variety of 2-part mixture problems.<br><BR>\n" );
document.write( "I'll mix the 30% and 50% solutions first.<br><BR>\n" );
document.write( "Using 3 times as much 50% solution as 30% solution means that, when I mix these two, 3/4 of the mixture will be the 50% acid.<BR>\n" );
document.write( "That means the percentage of the mixture will be 3/4 of the way from 30% to 50%.<BR>\n" );
document.write( "3/4 of the way from 30% to 50% is 45%.<br><BR>\n" );
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document.write( "NOTE: That's the key to using this method of solving 2-part mixture problems. 3/4 of this mixture being 50% acid and 1/4 being 30% acid means the percentage of the mixture will be 3/4 of the way from 30% to 50%.<br><BR>\n" );
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document.write( "So mixing the 30% and 50% acid results in a mixture that is 45% acid.<br><BR>\n" );
document.write( "Now the problem is mixing that 45% acid solution with the given 15% acid solution to obtain a mixture that is 35% acid.<BR>\n" );
document.write( "35% is 2/3 of the way from 15% to 45%<BR>\n" );
document.write( "So 2/3 of the final mixture is the 45% acid and 1/3 is the 15% acid.<br><BR>\n" );
document.write( "Now I'm ready to find the amounts of each.<br><BR>\n" );
document.write( "1/3 of the final mixture is the 15% acid.  That's 1/3 of 96 liters, or 32 liters.<br><BR>\n" );
document.write( "The other 2/3 of the mixture, or 64 liters, is the 45% acid, of which 3/4 is the original 50% acid and 1/4 is the original 30% acid.  That makes 48 liters of the 50% acid and 16 liters of the 30% acid.<br><BR>\n" );
document.write( "ANSWERS: 48 liters of 50% acid; 16 liters of 30% acid; 32 liters of 15% acid.<br><BR>\n" );
document.write( "CHECK:<BR>\n" );
document.write( ".50(48)+.30(16)+.15(32) = 24+4.8+.48 = 33.6<BR>\n" );
document.write( ".35(96) = 33.6<br><BR>\n" );
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