Question 499035
In work problems like this, you add rates of
working to get the rate working together
Let Mike's rate = ( 1 car ) / ( m minutes ) = {{{ 1/m }}}
Pat's rate = ( 1 car ) / ( m + 10 min ) = {{{ 1/( m + 10) }}}
Their rate working together = ( 1 car ) / ( 12 min ) = {{{ 1/12 }}}
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{{{ 1/m + 1/( m + 10 ) = 1/12 }}}
Multiply both sides by {{{ 12*m*( m + 10 ) }}}
{{{ 12*( m + 10 ) + 12m = m*( m + 10 ) }}}
{{{ 12m + 120 + 12m = m^2 + 10m }}}
{{{ m^2 - 14m - 120 = 0 }}}
I'll use the quadratic formula 
{{{m = ( -b +- sqrt( b^2 - 4*a*c )) / (2*a) }}} 
{{{ a = 1 }}}
{{{ b = -14 }}}
{{{ c = -120 }}}
{{{m = ( -(-14) +- sqrt( (-14)^2 - 4*1*(-120) )) / (2*1) }}} 
{{{m = ( 14 +- sqrt( 196 + 480 )) / 2 }}} 
{{{m = ( 14 +- sqrt( 676 )) / 2 }}} 
{{{ m = ( 14 + 26 ) / 2 }}} ( ignore negative root )
{{{ m = 20 }}}
It takes Mike 20 min working alone
check answer:
{{{ 1/m + 1/( m + 10 ) = 1/12 }}}
{{{ 1/20 + 1/( 20 + 10 ) = 1/12 }}}
{{{ 1/20 + 1/30 = 1/12 }}}
Multiply both sides by {{{60}}}
{{{ 3 + 2 = 5 }}}
OK