Question 971683
Add their rates of cleaning to get their
rate cleaning together
Let {{{ t }}} = dad's time in hrs  to clean house
working alone
{{{ t + 1 }}} = son's time in hrs to clean house
working alone
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In words:
[ 1 house cleaned ] / [ t hrs ] + [ 1 house cleaned ] / [ t + 1 hrs ] =
[ 1 house cleaned ] / [ 6 hrs ]
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{{{ 1/t + 1/( t + 1 ) = 1/6 }}}
Multip[ly both sides by {{{ t*( t+1 )*6 }}}
{{{ ( t+1 )*6 + 6t = t*( t+1 ) }}}
{{{ 6t + 6 + 6t = t^2 + t }}}
{{{ t^2 - 11t - 6 =  0 }}}
Use quadratic formula
{{{ t = ( -b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 1 }}}
{{{ b = -11 }}}
{{{ c= -6 }}}
{{{ t = ( -(-11) +- sqrt( (-11)^2 - 4*1*(-6) )) / (2*1) }}}
{{{ t = ( 11 +- sqrt( 121 + 24 )) / 2 }}}
{{{ t = ( 11 +- sqrt( 145 )) / 2 }}}
{{{ t = ( 11 + 12.042 ) / 2 }}}
{{{ t = 23.042 / 2 }}}
{{{ t = 11.521 }}}
and
{{{ t + 1 = 12.521 }}}
{{{ .521*60 = 31.26 }}}
Working alone, the son takes about 12 hrs 31 min
check:
{{{ 1/t + 1/( t + 1 ) = 1/6 }}}
{{{ 1/11.521 + 1/12.521 = 1/6 }}}
{{{ .0868 + .0799 = .16667 }}}
{{{ .1667 = .16667 }}}
close enough