document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204890>Question 1204890</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A scientist has two solutions, which she has labeled Solution A and Solution B. Each contains salt. She knows that Solution A is 45% salt and Solution B is 95% salt. She wants to obtain 90 ounces of a mixture that is 65% salt. How many ounces of each solution should she use? </SPAN>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "<font color=red>54 ounces</font> of solution A<BR>\n" );
document.write( "<font color=red>36 ounces</font> of solution B\r<BR>\n" );
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document.write( "Work Shown\r<BR>\n" );
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document.write( "Method 1\r<BR>\n" );
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document.write( "A+B = 90<BR>\n" );
document.write( "B = 90-A<BR>\n" );
document.write( "0.45A = amount of pure salt from solution A<BR>\n" );
document.write( "0.95B = amount of pure salt from solution B<BR>\n" );
document.write( "0.45A+0.95B = total amount of pure salt<BR>\n" );
document.write( "0.45A+0.95B = 90*0.65<BR>\n" );
document.write( "0.45A+0.95(90-A) = 58.5<BR>\n" );
document.write( "0.45A+0.95*90+0.95*(-A) = 58.5<BR>\n" );
document.write( "0.45A+85.5-0.95A = 58.5<BR>\n" );
document.write( "-0.5A + 85.5 = 58.5<BR>\n" );
document.write( "-0.5A = 58.5 - 85.5<BR>\n" );
document.write( "-0.5A = -27<BR>\n" );
document.write( "A = -27/(-0.5)<BR>\n" );
document.write( "A = <font color=red>54</font><BR>\n" );
document.write( "B = 90-A = 90-54 = <font color=red>36</font>\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "Solution A: 45%<BR>\n" );
document.write( "Solution B: 95%<BR>\n" );
document.write( "Target: 65%\r<BR>\n" );
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document.write( "The gap from 45% to 65% is 20<BR>\n" );
document.write( "The gap from 65% to 95% is 30<BR>\n" );
document.write( "The ratio 20:30 reduces to 2:3\r<BR>\n" );
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document.write( "We need 2 parts of one solution and 3 parts of another solution.<BR>\n" );
document.write( "That scales up to 2x and 3x.<BR>\n" );
document.write( "2x+3x = total amount of mixed solution<BR>\n" );
document.write( "2x+3x = 90 <BR>\n" );
document.write( "5x = 90<BR>\n" );
document.write( "x = 90/5<BR>\n" );
document.write( "x = 18\r<BR>\n" );
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document.write( "2x = 2*18 = <font color=red>36</font><BR>\n" );
document.write( "3x = 3*18 = <font color=red>54</font><BR>\n" );
document.write( "We know from method 1 which value goes where, but if we didn't have that luxury then we'd have to check each case.\r<BR>\n" );
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document.write( "Case 1: We have 54 oz of A and 36 oz of B<BR>\n" );
document.write( "Case 2: We have 36 oz of A and 54 oz of B\r<BR>\n" );
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document.write( "Case 1 leads to 54*0.45 + 36*0.95 = 58.5 oz of pure salt, which matches with the result of 90*0.65 = 58.5<BR>\n" );
document.write( "This confirms the answers found in method 1.\r<BR>\n" );
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document.write( "Case 2 yields 36*0.45+54*0.95 = 67.5 oz of pure sault which doesn't match 90*0.65 = 58.5<BR>\n" );
document.write( "We can eliminate this option.<BR>\n" );
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