A person travels a certain distance at 3 km/h and reaches 15 min late. If he travels at 4 km/h, he reaches 15 min earlier. The distance he has to travel is.?

Relative speed = 21 - 15 = 6 m per minute.
Relative distance = 114 m
Time taken to catche = 144/6 min = 19 min

Required average speed = Total distance covered/Total time taken
= (3 x 39)/[39/26 + 39/39 +39/52]
= 3 x 13/(1/2 + 1/3 + 1/4)
= 3 x 13 x 12/13 = 36 km/h

Let they meet after T h.
then, according to the question, 3T + 4T = 17.5
? 7T = 17.5
? T = 17.5/7 = 2.5 h
So, they meet 2.5 h after 8 : 00 am.
It means they meet at 10 : 30 am.

Let usual speed of a man is V and time is t h.
By the formula, V₁t₁ = V₂t₂
Vt = 1¹/₂ V(t - 20/60)
? t = 3/2(t - 1/3)
? 3/2 t - t = 1/2
? t = 1 h
Hence, the time taken by the man to reach his office at his usual speed is 1 h.

Let the total distance covered = L = Length of journey
According to the question,
(L/2 x 1/40) + (L/2 x 1/60) = 16
? L/80 + L/120 = 16
? (3L + 2L)/240 = 16
? 5L = 16 x 240
? L = (16 x 240)/5 = 16 x 48 = 768 km

Let distance and original speed of the man be d km and s km/h.
Then,
d/s - d/s + 3 = 2/3
? [d(s + 3 - s)]/[s(s + 3)] = 2/3
? 9d = 2s(s + 3) ...(i)

And d/(s - 2) - d/s = 2/3
? [d(s - s + 2)]/[s(s - 2)] = 2/3
? 3d = s(s - 2) ...(ii)

From Eqs.(i) and (ii), we get
3s(s - 2) = 2s(s + 3)
? 3s² - 6s = 2s² + 6s
? s² = 12s
? s = 12

From Eq.(ii), we get
3d = 12(12 - 2)
? d = 40 km

Here, a = 10 km/h
b = ( 10 + 2 ) = 12 km/h,
t₁ = 6 min = 1/10 h
and t₂= 1/10 h.
? Required distance = ab( t₁ + t₂ )/(b - a)
= 10 x 12 x ( 1/10 + 1/10 ) /(12 - 10) = (10 x 12 x 1)/(5 x 2) = 12km

Let B catches A after T h.
Then, distance travelled by A in (T + 2) h = Distance travelled by B in T h

According to the question,
2 (T + 2 ) = 5T
? 2T + 4 = 5T
? 3T = 4
? T = 4/3 h

Distance travelled by B in 4/3 h
= (4/3) x 5 = 20/3 = 6²/₃ km

Let total distance be 100 km.
Then, Average = 100/[ (50/40) + (30/60) + (20/30)]
= 100/( 5/4 + 1/2 + 2/3) = 100/[(15 + 6 + 8)/12]
= (100 x 12)/29
= 42.35

Distance of his office from his house = {ab/(b - a)} x (t₁ + t₂)
Here, a = 20 km/h, t₁ = 1 h, b = 20 + 5 = 25 km/h
and t₂ = 30 min = 0.5 h
? Distance = {20 x 25/(25 - 20)} x (1 + 0.5) = 500(1.5)/5
= 100 x 1.5 = 150 km