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Type 1:

When you are given two different speeds (s1 and s2) for travelling through a certain distance, and total time (t) for these two journeys:

Formula:

                                          Distance = (S1*S2)/[(S1+S2)t]

Ex 1: A boy/train goes from A to B at 3 km/hr, back from B to A at 2 km/hr. Total time for these two journeys is 5 hours, then distance from A to B is given by:
Sol:
Distance = Product of speeds/Addition of speeds * Time
D = 3*2/(3+2) * 5 = 6 km
D = 6km
Ex 2: A boy/train travels from A to B. He covers half distance at 3 km/hr, and other half at 2 km/hr. Total time taken is 5 hours, then distance from A to B is given by:
Sol:
So this time total distance from A to B = (2* Product of speeds)/(Addition of speeds * Time)
D = 2 * [3*2/(3+2) * 5 ]= 12 km
D = 12km
Ex 3: A boy/train travels from A to B. He covers 2/3rd distance at 2 km/hr, and rest at 3 km/hr. Total time taken is 24 hours, then distance from A to B is given by:
Sol:
S1 = 2 km/hr, S2 = 3 km/hr, Total time, T = 24 hrs
Distance left = 1 – 2/3 = 1/3
Reciprocate both distances, i.e. D1 = 3/2 and other D2 = 3/1
Distance from A to B = D1*S1 * D2*S2/ (D1*S1 + D2*S2) * T
i.e. 3/2*2 * 3/1*3/ (3/2*2 + 3/1*2) * 24 = 54 km
D = 54km
Type 2:
When you are given two different speeds (s1 and s2) for travelling through a certain distance, and total difference in time (t) is given for these two journeys:

Formula:

                                        Distance = (S1*S2)/[(S1-S2)t]

Ex 1: A boy/train goes from A to B. If speed is 30 km/hr, he/it is late by 10 minutes. If speed is 40 km/hr, he/it reaches 5 minutes earlier, then distance from A to B is given by:
Sol:
Difference in time = 10 – (-5) = 15 minutes or 1/4 hr……..earlier(-5)
Distance = Product of speeds/Difference of speeds * Difference in time
= 30*40/(40-30) * 1/4 = 30 km
D = 30km
Ex 2: A train leaves Delhi at 9 AM at 25 km/hr, another train at 35 km/hr leaves Delhi at 2 PM in same direction. How many kilometres from Delhi they will be together?
Sol:
Difference in times = 2 PM – 9 AM = 5 hrs
Distance from Delhi = 25*35/(35-25) * 5
D = 17.5km
Type 3:
When different speed and time are given to find the original time.
Formula:
Original time = (Reciprocal of given speed*t) – t(late)
Original time = (Reciprocal of given speed*t) + t(early)
Ex 1: Walking at 3/4th of usual speed, person is late by 10 minutes. Find the usual time.
Sol:
Let t be usual time.
Reciprocate speed for time, i.e. 4/3 * t
Person is late by 10 minutes, so (4/3 * t) – t = 10
t = 30 min
Type 4:To find the time of halts when the speed with stoppages (s1)and speed without stoppages (s2) is given.
Formula:
Halt time = (s2-s1)/s2
Ex 1: A man/train leaves from A to B. Speed with stoppages = 60 km/hr, speed without stoppages = 80 km/hr. Find the time man/train stop?
Sol:
Minutes per hour the man/ train stops = (80-60)/80 = 1/4 hr = 15 min
Time = 15 min
Type 5 :
A & B start at same time towards each other. After meeting each other, they reach their destinations after ‘a’ and ‘b’ hrs respectively.

Formula:

                                           A’s Speed : B’s Speed = √b : √a

Ex 1: Two, trains, one from Delhi to Mumbai and the other from Mumbai to Delhi, start simultaneously. After they meet, the trains reach their destinations after 4 hours and 9 hours respectively. The ratio of their speeds is:
Sol:
A:B = √b : √a = 3 : 2
A:B = 3:2
Type 6 :
A man takes ‘a’ hrs to walk to a certain distance and ride back.
  1. If riding both ways takes him b hrs, then Walking both ways takes him
                                                  [a + (a-b)] hrs
  2. If walking both ways takes him b hrs, then Riding both ways takes him
                                                  [a – (b-a)] hrs
Ex 1: A man takes 6 hrs to walk to a certain distance and ride back. If riding both ways takes him 4 hrs, then Walking both ways takes him how much time?
Sol:
T = [a + (a-b)] hrs
T = 6 +2
T = 8hrs
Ex 2: A man takes 5 hrs to walk to a certain distance and ride back. If walking both ways takes him 7 hrs, then Riding both ways takes him how much time?
Sol:
T = [a – (b-a)] hrs
T = 5 – 2
T = 3hrs
Type 7 :
A man/train leaves from A at x1 AM and reaches Q at y1 AM. Another man/train leaves from Q at x2 AM and reaches P at y2 PM. When they will meet?

Formula:
                                         They will meet at = (x1+y2)/2
Ex 1: A man/train leaves from P at 6 AM and reaches Q at 10 AM. Another man/train leaves from Q at 8 AM and reaches P at 12 PM. When they will meet
Sol:
Since both takes 4 hrs to reach their destination, so they will exactly at the middle of 6 AM and 12 PM, i.e. at 9 AM.

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