document.write( "Question 1019704: Two grain-collecting machines can collect all the grain from a field 9 days faster than if the first one was doing it alone and 4 days faster than if the second one was working alone. How long does it take each grain-collecting machine to collect all the grain by itself? \n" ); document.write( "

Algebra.Com's Answer #635691 by ikleyn(52911) $\"\"$ $\"About$

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\n" ); document.write( "Two grain-collecting machines can collect all the grain from a field 9 days faster than if the first one was doing it alone,
\n" ); document.write( "and 4 days faster than if the second one was working alone. How long does it take each grain-collecting machine
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document.write( "Let f be the number of days for the first  grain-collecting machines to collect all the grain from a field, if it works alone.\r\n" );
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document.write( "Let s be the number of days for the second grain-collecting machines to collect all the grain from a field, if it works alone.\r\n" );
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document.write( "Then  the rate of the first  machine is  of the \"field-per-day\", \r\n" );
document.write( "while the rate of the second machine is  of the \"field-per-day\".\r\n" );
document.write( "When both machines work simultaneously, their combined rate is  of the \"field-per-day\".\r\n" );
document.write( "Hence, two machines working together can can collect all the grain from the field in  days.\r\n" );
document.write( "Therefore, from the first part of the condition you have this equation:\r\n" );
document.write( "f -  = 9.   (1)\r\n" );
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document.write( "From the second part of the condition you have this equation:\r\n" );
document.write( "s -  = 4.   (2)\r\n" );
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document.write( "Thus you have the system of two non-linear equations (1) and (2). \r\n" );
document.write( "Do not be scared: still there is a way to solve it.\r\n" );
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document.write( "Distract equation (2) (both sides) from equation (1). You will get\r\n" );
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document.write( "f - s = 5,   or   f = s + 5.   (3)\r\n" );
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document.write( "Now substitute the expression  f = s + 5  from (3) into equation (1). You will get a single equation for the unknown s:\r\n" );
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document.write( "(s+5) -  = 9.\r\n" );
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document.write( "Simplify and solve it, step by step:\r\n" );
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document.write( "(s+5) -  = 9,  --->  s+5 -  = 9,  --->  . . .  --->   = .\r\n" );
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document.write( "To solve the last quadratic equation, factor it:  = (s-10)*(s+2),\r\n" );
document.write( "and you will get the unique solution to the problem  s = 10.  (The other root is negative and, therefore, doesn't suit).\r\n" );
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document.write( "Answer. It will take 10 days for the second machine to complete the job working alone.\r\n" );
document.write( "        It will take 15 = 10+5 days for the first  machine to complete the job working alone.\r\n" );
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