Question 942720
Let {{{ s }}} = Hector's speed in mi/hr
{{{ s + 2 }}} = Maureen's speed in mi/hr
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Hector's head start is:
{{{ d[1] = s*(1/2) }}}
{{{ d[1] = .5s }}}
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Start a stopwatch when Maureen leaves
Let {{{ t }}} = time in hrs on stopwatch 
when she catches Hector
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Hector's equation:
(1) {{{ 12 - .5s = s*t }}}
Maureen's equation:
(2) {{{ 12 = ( s+2 )*t }}}
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(1) {{{ s*t + .5s = 12 }}}
(1) {{{ s*( t + .5 ) = 12 }}}
(1) {{{ s = 12 / ( t + .5 ) }}}
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(2) {{{ 12 = s*t + 2t }}}
(2) {{{ s*t = 12 - 2t }}}
(2) {{{ s = ( 12 - 2t ) / t }}}
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By substitution:
{{{ 12/( t + .5 ) = ( 12 - 2t ) / t }}}
{{{ 12t = ( 12 - 2t )*( .5 + t ) }}}
{{{ 12t = 6 - t + 12t - 2t^2 }}}
{{{ 2t^2 + t - 6 = 0 }}}
{{{ ( 2t - 3 )*( t + 2 ) = 0 }}} ( just a wild guess )
{{{ 2t = 3 }}}
{{{ t = 1.5 }}} ( can't have negative time )
(2) {{{ 12 = ( s+2 )*t }}}
(2) {{{ 12 = ( s+2 )*1.5 }}}
(2) {{{ 12 = 1.5s + 3 }}}
(2) {{{ 1.5s = 9 }}}
(2) {{{ s = 6 }}}
and
{{{ s + 2 = 8 }}}
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Hector runs at 6 mi/hr
Maureen runs at 8 mi/hr
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check:
(1) {{{ 12 - .5s = s*t }}}
(1) {{{ 12 - .5*6 = 6*1.5 }}}
(1) {{{ 12 - 3 = 9 }}}
(1) {{{ 9 = 9 }}}
OK