Question 824509
If you add their rates of working, you
will get their rate of working together
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Cody's rate is ( 1 job ) / ( 8 hrs )
Kaitlin's rate is ( 1 job ) / ( 6 hrs )
Let Joseph's rate = ( 1 job ) / ( t hrs )
Let their rate working together = ( 1 job ) / ( T hrs )
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I can say:
{{{ 1/8 + 1/6 + 1/t = 1/T }}}
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The time, {{{ T }}} has to be less than {{{ 6 }}} hrs
no matter how slow Joseph is.
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Suppose Joseph is the slowest and takes 
{{{ 12 }}} hrs to do the job alone
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{{{ 1/8 + 1/6 + 1/t = 1/T }}}
{{{ 1/8 + 1/6 + 1/12 = 1/T }}}
Multiply both sides by {{{ 24T }}}
{{{ 3T + 4T + 2T = 24 }}}
{{{ 9T = 24 }}}
{{{ T = 2.667 }}} hrs
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If Joseph happens to be faster than the other two,
the job will get done in LESS than Joseph's time
Example:
Joseph's rate is {{{ 1/2 }}}
{{{ 1/8 + 1/6 + 1/t = 1/T }}}
{{{ 1/8 + 1/6 + 1/2 = 1/T }}} 
{{{ 3T + 4T + 12T = 24 }}}
{{{ 19T = 24 }}}
{{{ T = 1.2632 }}} hrs