Question 634130
Since the length of the track isn't given,
I'll say it is {{{ 40 }}} units.
So the faster train's speed is {{{ 40 / 40 = 1 }}} unit/sec
The slower train's speed is {{{ 40/55 }}} units/sec
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I think of the round track as straightened out 
with a mark on the track every {{{ 40 }}} units.
Then the problem becomes:
"What is the smallest time {{{ t }}} when they both 
end up on a mark?" The mark is really the same point
repeated.
The faster train:
(1) {{{ j*40 = 1*t }}}
The slower train:
(2) {{{ k*40 = (40/55)*t }}}
( j and k are whole numbers )
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Solving for {{{t}}},
(1) {{{ t = 40j }}}
(2) {{{ t = 55k }}}
so, {{{ 40j = 55k }}}
{{{ j/k = 55/40 }}}
{{{ j/k = 11/8 }}}
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This tells me the faster train goes around 11 times,
and the slower train goes around 8 times in time
{{{ t }}} , meeting back at the start
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(1) {{{ t = 40j }}}
(1) {{{ t = 40*11 }}}
(1) {{{ t = 440 }}} sec
and, checking
(2) {{{ t = 55k }}}
(2) {{{ t = 55*8 }}}
(2) {{{ t = 440 }}} sec