document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1145132>Question 1145132</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a 10-quart container is filled with water.One quart of water is drained out and replaced with alcohol. After mixing a quart of the solution is drainec out and replaced with alcohol. This process is continued until 5 quarts of alvohol have been put into the container. The solution in the container is then what percent of alcohol. </SPAN>\n" );
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document.write( "At the start, the 10 quarts are 10 of water and 0 of alcohol.<br><BR>\n" );
document.write( "(1) First time:  When 1/10 (0.1) of the container is emptied, the amount of alcohol removed is (0.1)(0)=0, leaving 0; when 1 quart of alcohol is then added, the amount of alcohol is 0+1 = 1.<br><BR>\n" );
document.write( "(2) Second time:  When 1/10 (0.1) of the container is emptied, the amount of alcohol removed is (0.1)(1)=0.1, leaving 0.9; when 1 quart of alcohol is then added, the amount of alcohol is 0.9+1 = 1.9<br><BR>\n" );
document.write( "(3) Third time:  When 1/10 (0.1) of the container is emptied, the amount of alcohol removed is (0.1)(1.9)=0.19, leaving 1.71; when 1 quart of alcohol is then added, the amount of alcohol is 1.71+1 = 2.71<br><BR>\n" );
document.write( "(4) Fourth time:  When 1/10 (0.1) of the container is emptied, the amount of alcohol removed is (0.1)(2.71)=0.271, leaving 2.439; when 1 quart of alcohol is then added, the amount of alcohol is 2.439+1 = 3.439<br><BR>\n" );
document.write( "(5) Fifth time:  When 1/10 (0.1) of the container is emptied, the amount of alcohol removed is (0.1)(3.439)=0.3439, leaving 3.0951; when 1 quart of alcohol is then added, the amount of alcohol is 3.0951+1 = 4.0951<br><BR>\n" );
document.write( "After the 5th time, the percentage of alcohol in the container is 4.0951/10 converted to a percentage, which is 40.951%.<br><BR>\n" );
document.write( "ANSWER: 40.951%<br><BR>\n" );
document.write( "And now that we have obtained the answer by that method, there is a much easier path....<br><BR>\n" );
document.write( "Each time a quart of liquid is removed and replaced with a quart of alcohol, the fraction of water in the container gets multiplied by 0.9.<BR>\n" );
document.write( "After 5 times, the fraction of the liquid in the container that is water is (0.9)^5 = 0.59049.<BR>\n" );
document.write( "So after the 5th time the percentage of water in the container is 59.049%; so the percentage of alcohol is 100-59.049% = 40.951%.<BR>\n" );
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