Question 597625
Add their rates of soaking to get their 
rate of soaking together
Let {{{ t }}} = the time in minutes for the larger sprinkler to soak lawn
{{{ t + 16 }}} is the time in minutes  for the smaller sprinkler to soak lawn
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Rate for larger sprinkler = ( 1 lawn ) / ( t min )\
Rate for smaller sprinkler = ( 1 lawn ) / ( t + 16 min )
Rate working together = ( 1 lawn ) / ( 6 min )
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{{{ 1/t + 1/( t + 16 ) = 1/6 }}}
Multiply each sides by {{{ 6t*( t + 16 ) }}}
{{{ 6*( t + 16 ) + 6t = t*( t + 16 ) }}}
{{{ 6t + 96 + 6t = t^2 + 16t }}}
{{{ t^2 + 4t = 96 }}}
Complete the square
{{{ t^2 + 4t + (4/2)^2 = 96 + (4/2)^2 }}}
{{{ t^2 + 4t + 4 = 96 + 4 }}}
{{{ ( t + 2 )^2 = 10^2 }}}
Take the square root of both sides
{{{ t + 2 = 10 }}} ( ignore the negative square root of {{{100}}} )
{{{ t = 8 }}}
and
{{{ t + 16 = 24 }}}
The smaller sprinkler takes 24 min
The larger one takes 8  min
check:
{{{ 1/t + 1/( t + 16 ) = 1/6 }}}
{{{ 1/8 + 1/24 = 1/6 }}}
{{{ 3/24 + 1/24 = 4/24 }}}
{{{ 4/24 = 4/24 }}}
OK