Kindly help me with this really tough problem that even my Maths or Physics teacher couldn't do

It's one of those challenging math problems, taken from a list of questions in an international mathematics competition, and I, for the life of me, can't do it. I know the answer, but kindly explain thoroughly how you arrived at your answer if you attempt this question. The question is:
Henry must travel from A to B and he plans to go at a certain speed. He would like to arrive earlier than planned and notes that travelling at a speed 5 km/h faster than planned he will arrive 5 hours earlier and travelling at a speed 10 km/h faster than planned he will arrive 8 hours earlier. What is his planned speed?

let planned speed = v km/h and planned time, t hrs, then

(v+5)(t-5) = vt ....... (i)

(v+10)(t-8) = vt ....... (ii)

solve to get
v = 15 km/h, t = 20 hrs

check:
---------
vt = 15*20 =300 km
(v+5)(t-5) = 20*15 = 300 km
(v+10)(t-8) = 25*12 = 300 km

distance = velocity * time

d = vt = (v+5)(t-5) = (v+10)(t-8)
d = vt = vt - 5v + 5t - 25 = vt - 8v + 10t - 80
so t = v+5 and 10t = 8(v+10) so
10v + 50 = 8v + 80
2v = 30
v = 15

Henry's planned speed was 15 km/h over 20 hours to cover 300km.
At 20 km/h that is 15 hours, and at 25 km/h that is 12 hours.

Hi,

(t - 5)(r + 5) = tr

(t - 8)(r + 10) = tr

tr + 5t - 5r - 25 = tr
tr +10t - 8r - 80 = tr

5t - 5r = 25
10t - 8r = 80

-2( 5t - 5r = 25)
10t - 8r = 80

-10t + 10r = -50
10t - 8r = 80
-------------------------
2r = 30
r = 15
t = 20

His planned speed is 15 km.h <==ANSWER

I hope that helps!! :-)

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