Question 755004
Add their separate rates to get their rate filling together
B's rate of filling:
( 1 pool filled ) / ( t hrs )
A's rate of filling:
( 1 pool filled ) / ( t - 3 hrs )
A and B together:
( 1 pool filled ) / ( 2 hrs )
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{{{ 1/( t - 3 ) + 1/t = 1/2 }}}
Multiply both sides by {{{ t*( t-3 )*2 }}}
{{{ 2t + 2*( t-3 ) = t*( t-3 ) }}}
{{{ 2t + 2t - 6 = t^2 - 3t }}}
{{{ 4t - 6 = t^2 - 3t }}}
{{{ t^2 - 7t + 6 = 0 }}}
{{{ ( t - 6 )*( t - 1 ) = 0 }}} ( by inspection )
{{{ t = 6 }}} ( note that {{{ t=1 }}} doesn't work )
{{{ t - 3 = 3 }}}
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A takes 3 hrs to fill the pool alone
B takes 6 hrs to fill the pool alone
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check:
{{{ 1/( t - 3 ) + 1/t = 1/2 }}}
{{{ 1/3 + 1/6 = 1/2 }}}
{{{ 2/6 + 1/6 = 3/6 }}}
{{{ 3/6 = 3/6 }}}
OK