Question 899314
Add their rates of painting:
Mike's rate:
( 1 room ) / ( 75 min )
Joan's rate:
( 1 room ) / ( 60 min )
Kyle's rate:
( 1 room ) / ( 80 min )
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Let {{{ t }}} = time in minutes with 
all 3 painting:
{{{ 1/75 +1/60 + 1/80 = 1/t }}}
Multiply both sides by {{{ 5 }}}
{{{ 1/15 + 1/12 + 1/16 = 5/t }}}
Multiply both sides by {{{ 240t }}}
{{{ 16t + 20t + 15t = 1200 }}}
{{{ 51t = 1200 }}}
{{{ t = 23.53 }}} 
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with all 3 painting it will take 23.53 min
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check:
{{{ 1/75 +1/60 + 1/80 = 1/t }}}
{{{ 1/75 +1/60 + 1/80 = 1/23.53 }}}
{{{ .01333 + .01666 + .0125 = .0425 }}}
{{{ .04249 = .0425 }}}
close enough