Question 945299
Let {{{ t }}} = Jen's time in minutes to shelve a cart
{{{ t - 14 }}} = Pat's time in minutes to shelve a cart
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Jen's rate:
( 1 cart ) / ( t min )
Pat's rate:
( 1 cart ) / ( t - 14 min )
Working together:
( 1 cart ) / ( 24 min )
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Add their rates to get rate working together
{{{ 1/t + 1/( t - 14 ) = 1/24 }}}
Multiply both sides by {{{ t*( t - 14 )*24 }}}
{{{ 24*( t - 14 ) + 24t = t*( t - 14 ) }}}
{{{ 24t - 336 + 24t = t^2 - 14t }}}
{{{ t^2 - 62t + 336 }}}
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Use quadratic formula
{{{ t = ( -b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 1 }}}
{{{ b = -62 }}}
{{{ c = 336 }}}
{{{ t = ( -(-62) +- sqrt( (-62)^2 - 4*1*336 )) / (2*1) }}}
{{{ t = ( 62 +- sqrt( 3844 - 1344 )) / 2 }}}
{{{ t = ( 62 +- sqrt( 2500 )) / 2 }}}
{{{ t = ( 62- 50 ) / 2 }}}
{{{ t = 6 }}} 
( this answer makes no sense, because it is faster than
their time working together- 24 min )
{{{ t = ( 62 + 50 ) / 2 }}}
{{{ t = 112/2 }}}
{{{ t = 56 }}} 
Working alone, Jen's time is 56 sec
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check:
{{{ 1/t + 1/( t - 14 ) = 1/24 }}}
{{{ 1/56 + 1/42 = 1/24 }}}
{{{ 1/28 + 1/21 = 1/12 }}}
{{{ 252/7056 + 336/7056 = 588/7056 }}}
{{{ 588 = 588 }}}
OK