Usuall peter & paula meet at a bus interchange where busses depart at equal periods of time.?

Question

once paula came 15mins later and peter saw 6 buses depart. the 2nd time paula came 26 mins later and peter sadly watched 8 buses depart. but the next time peter came 43mins later than paula. How many buses departed the interchange while Paula was waiting for Peter?
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plz help peoples im baffled by this question

Answer 1

Suppose buses leave every x minutes. If you see N buses depart, this may take any number of minutes from (N-1)x [if the first bus departs just after you arrive and the last one departs just before you leave] to (N+1)x [if a bus departed just before you arrive and another one departs just after you leave].

So we know:
5x ≤ 15 ≤ 7x
7x ≤ 26 ≤ 9x
i.e.
x ≤ 3
x ≥ 15/7 = 2 1/7
x ≤ 26/7 = 3 5/7
x ≥ 26/9 = 2 8/9
In summary, 26/9 ≤ x ≤ 3. So x is very close to 3. ;-)

If x = 26/9, 43 minutes = 14.9 x; if x = 3, 43 minutes = 14.3 x. It must be between these, so it's between 14x and 15x, so there could be either 14 or 15 buses departing in this time period.

Answer 2

let b be the time between two buses.
on the first day the first bus arrived x mins after peter came and the last bus came y mins before paula arrived
so on the first day x+y+6b=15 where 0 on the second day the first bus arrived p mins after peter came and the last bus came q mins before paula arrived
so on the second day p+q+8b=26 where 0 similarly on the 3rd day s+t+nb=43 where 0 and n is to be found
subtracting the first equation from the second
p+q-x-y+2b=11
-2b adding 2b in the above
0 0<11<4b
got stuck after this