Question 883050
Let {{{ t }}} = time in hours it takes
Tim to paint the room
{{{ t + 4 }}} = time in hours for
Ronnie to paint the room
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Add their rates of working to get
their rate working together
{{{ 1/t + 1/( t+4) = 1/9 }}}
Multiply both sides by {{{ 9*t*( t+4 ) }}}
{{{ 9*( t+4 ) + 9t = t*( t+4 ) }}}
{{{ 9t + 36 + 9t = t^2 + 4t }}}
{{{ t^2 - 14t = 36 }}}
Complete the square
{{{ t^2 - 14t + (14/2)^2 = 36 + (14/2)^2 }}}
{{{ t^2 - 14t + 49 = 36 + 49 }}}
{{{ ( t - 7 )^2 = 85 }}}
{{{ t - 7 = sqrt(85) }}}
{{{ t = 7 + sqrt(85) }}} ( Tim's time )
{{{ t + 4 = 11 + sqrt(85) }}} ( Ronnie's time )
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check:
{{{ 1/t + 1/( t+4) = 1/9 }}}
{{{ 1/( 7 + sqrt(85)) + 1/( 11 + sqrt(85)) = 1/9 }}}
{{{ ( 11 + sqrt(85) )*9 + ( 7 + sqrt(85) )*9 = ( 7 + sqrt(85) )*( 11 + sqrt(85) ) }}}
{{{ 99 + 9*sqrt(85) + 63 + 9*sqrt(85) = 77 + 18*sqrt(85) + 85 }}}
{{{ 99 + 63 = 77 + 85 }}}
{{{ 162 = 162 }}}
OK
Hope I got it