document.write( "Question 167380: A and B can do a piece of work in 42 days, B and C in 31 days, and A and C in 20 days. Working together, how many days can all of them finish the work? \n" ); document.write( "

Algebra.Com's Answer #410536 by Edwin Parker(36) $\"\"$ $\"About$

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A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together?\r
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\n" ); document.write( "\n" ); document.write( "A and B can do a piece of work in 42 days,
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document.write( "So A's and B's combined rate is 1 job per 42 days or  or \r\n" );
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\n" ); document.write( "B and C in 31 days
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document.write( "So B's and C's combined rate is 1 job per 31 days or  or \r\n" );
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\n" ); document.write( "C and A in 20 days
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document.write( "So C's and A's combined rate is 1 job per 31 days or  or \r\n" );
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document.write( "Suppose A's rate working alone is is 1 job per x days or  or .\r\n" );
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document.write( "Suppose B's rate working alone is is 1 job per y days or  or .\r\n" );
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document.write( "Suppose C's rate working alone is is 1 job per z days or  or .\r\n" );
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document.write( "Suppose their combined rate is 1 job per d days or  or .\r\n" );
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document.write( "The four equations come from:\r\n" );
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document.write( "       +  = \r\n" );
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document.write( "             +  = \r\n" );
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document.write( "       +  = \r\n" );
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document.write( "             +  = \r\n" );
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document.write( "       +  = \r\n" );
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document.write( "             +  = \r\n" );
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document.write( "       +  +  = \r\n" );
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document.write( "             +  +  = \r\n" );
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document.write( " +  = \r\n" );
document.write( " +  = \r\n" );
document.write( " +  = \r\n" );
document.write( " +  +  = \r\n" );
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document.write( "Now we must find their combined rate which is\r\n" );
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document.write( "So we line up the first three equations like this and add them all:\r\n" );
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document.write( "           +          = \r\n" );
document.write( "               +      = \r\n" );
document.write( "               +      = \r\n" );
document.write( "         ---------------------\r\n" );
document.write( "           +  +      = \r\n" );
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document.write( "           +  +      = \r\n" );
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document.write( "Dividing both side by 2\r\n" );
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document.write( "           +  +      = \r\n" );
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document.write( "And since the fourth equation is\r\n" );
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document.write( "           +  +  = \r\n" );
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document.write( "Since things equal to the same thing are equal to each other,\r\n" );
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document.write( "            = \r\n" );
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document.write( "Cross-multiplying:\r\n" );
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document.write( "       1381d = 26040\r\n" );
document.write( "           d = \r\n" );
document.write( "           d = 18.85590152 days\r\n" );
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document.write( "Edwin

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