Question 990269
Add their rates of cleaning to get rate
working together
[ 1 garage cleaned ] / [ t hrs ] = Sarah's rate
[ 1 garage cleaned ] / [ t + 9 hrs ] = Heidi's rate
[ 1 garage cleaned ] / [ 6 hrs ] = their rate working together
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{{{ 1/t + 1/( t+9 ) = 1/6 }}}
Multiply both sides by {{{ 6t*( t+9) }}}
{{{ 6*( t + 9 ) + 6t = t*( t+ 9 ) }}}
{{{ 6t + 54 + 6t = t^2 + 9t }}}
{{{ 12t + 54 = t^2 + 9t }}}
{{{ t^2 - 3t - 54 = 0 }}}
I notice that {{{ 6*9 = 54 }}}
and {{{ 6 - 9 = -3 }}}, so
{{{ ( t - 9 )*( t + 6 ) = 0 }}}
{{{ t = 9 }}} ( can't have {{{ t = -6 }}}, time can't be negative )
and
{{{ t + 9 = 18 }}}
It will take Heidi 18 hrs to clean garage working alone
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check:
{{{ 1/t + 1/( t+9 ) = 1/6 }}}
{{{ 1/9 + 1/18 = 1/6 }}}
{{{ 2/18 + 1/18 = 3/18 }}}
{{{ 3/18 = 3/18 }}}
OK