Total time taken by a person in going to a place by walking and returning on cycle is 5 h 45 min. He would have gained 2 h by cycling both ways. The time taken by him to walk both ways, is?

Let T h, he travels on foot.
? 4T + 10(8 - T) = 50
? 80 - 6T = 50
? 6T = 30,
? T = 5 h
? The distance travelled on foot = 4 x 5 = 20 km

Relative speed = (58 - 32) = 26 km/h
= (26 x 5/18) m/s = 65/9 m/s
Time taken to cross each other = (250 + 140)/(65/9)
= (390 x 9)/65 = 54 s

Let the distance be x km.
? Time = Distance/speed
1st day time taken = x/4 h
2nd day time taken = x/6 h
Difference in time = (x/4 - x/6) h
Actual difference between these two times = 15 min = 1/4 h
? x/4 - x/6 = 1/4 ? (3x - 2x)/12 = 1/4
? x = 3 km

Time taken by express train to cover 75 km including stoppage
= (60 / 100 x 75) min + = 48 min
Time taken by express train to cover 600 km.
= Time taken by it to cover 75 km
= (48 / 75 x 525 ) min + (60 / 100 x 75 ) min
= (336 + 45) min = 381
Time taken by local train to9 cover 25 km including stoppage
= (60 / 50 x 25) min + 1 min = 31 min
In 31 min, distance covered = 25 km
In(31 x 12) min. distance covered
= (25 / 31 x 31 x 12 ) = 300 km
In last 9 min, distance covered
= (25 / 31 x 9 ) = 7.25 km
? Required total distance
= 300 + 7.25 = 307.25 km

Required number of hours is the number of terms of the series 40 + 45 + 50 +... as speed increases every hour.
Given, sum of the series is 385
? a = 40, d=5, s = 385 and n= ?
Using S = n/2[2a + (n - 1)d]
? 385 = n/2[80 + 5n - 5]
? 770 = 5n² + 75n
? n² + 15n - 154 = 0
? n² + 22n - 7n - 154 = 0
? n(n + 22) - 7(n + 22) = 0
? (n + 22) (n - 7) = 0
? n = 7
Remaining part = 1/12 - 1/20 = (5 - 3)/60 = 2/60 = 1/30 i.e., 1/30 th part is filled by B in 1 min
Hence, required time to fill the whole tank = (165 + 1 + 1) min = 167 min

Let w be the time taken in one way by walking and c be the time taken in one way by car.
Now, according to the question,
In first case, w + c = 6 h 45 min
or
2w + 2c = 13 h 30 min ..(i)
In second case, 2c = 4 h 45 min ....(ii)
From Eqs. (i) and (ii), we get
2w + 2c = 13 h 30 min
? 2w + 4h 45 min = 13 h 30 min
? 2w = 13 h 30 min - 4 h 45 min
? 2w = 8 h 45 min
? If he walks both ways, then time taken = 8 h 45 min

Since, the speed of X and Y are 60 km/h each. The distance between A and B is 700 km. Distance traveled by X upto 11 am is 60 km. Since, X stops at 10 am and distance traveled by up to 11 am is 120 km.
Now, the distance between them = 700 - 180 = 520 km
Now, let they will meet at a distance of x km from X's position.
? Distance traveled by X = x km
and distance traveled by Y = (520 - x) km
? x/60 = (520 - x)/60
? 2x = 520
? x = 260
? T = 260/60 = 4¹/₃ h
Thus, they will cross each other after, 4¹/₃ h after 11 am i.e., at 3 : 20 pm.

We know that, if two equal distances are covered at two different speeds A and B, then
Average speed = 2AB/(A + B) = (2 x 58 x 52)/(58 + 52)
= 54.8 km/h
= 55 km/h (Approx)

Distance between Sonagarh and Raipur = Average speed x time
= 69 x (35/60) km = 161/4 km
New speed = (69 + 36) km/h = 105 km/h
? Required time = Distance/speed
= 161/(4 x 105) h
= (161 x 60)/(4 x 105) min
= 23 min

Let the required distance = L
According to the question,
L/(5/2) - L/(7/2) = 12/60 (difference between two times = 6 + 6 =12 min)
? 2L/5 - 2L/7 = 1/5
? 14L - 10L = 7
? 4L = 7
? L = 7/4 km