Shortcut of Time, Speed and Distance-2

Shortcut 45:

Average speed when distances are same

                  S_a=\frac{2S_1S_2}{(S_1+S_2)}

Question:
Sai travels from her house to office at a speed of 50kmph. She suddenly returns home at a speed of 60kmph. What is her average speed?
Answer:
S1 = 50
S2 = 60
Sa = 2(50×60)/(50+60)
= 6000/110
= 54.54 kmph

Shortcut 46:
Average speed when time is same

     S_a=\frac{S_1+S_2}2

Question:
A man travels 400 km at 50kmph. He then travels another 320 km at 40kmph. What is the average speed?
Answer:
Time taken for 400km = 400/50 = 8 hours
Time taken for 320km = 320/40 = 8 hours
Since the time is same in both cases, use the above formula to find out the average speed.
S1 = 50
S2 = 40
Sa = (50+40)/2 = 45kmph

Shortcut 47:
Average speed when time or distance are not same

    S_a=\frac{D_1+D_2}{{\displaystyle\frac{D_1}{S_1}}+{\displaystyle\frac{D_2}{S_2}}}

Question:
A man travels 200 km at a speed 40kmph. He then travels 300 km at 50 kmph. Find his average speed.
Answer:
D1 = 200; D2 = 300
S1 =  40; S2 = 50
Substitute the values in the above equation, we get
Sa = (200+300)/[(200/40)+(300/50)]
Sa = 500/(5+6) = 45.45 kmph

Shortcut 48:
Least time to meet in a circular race

LCM(T_1,T_2,….)

T1, T2, T3, …. = Individual times taken by each racer to complete one circle.
Question:
Three persons participate in a race on a circular track of length 400m. They can run at a speed of 2mps, 4mps and 5mps respectively. How long will they take to meet in the starting point for the first time?
Answer:
Time taken by each person to complete one circle
Person 1 = 400/2 = 200 seconds = T1
Person 2 = 400/4 = 100 seconds = T2
Person 3 = 400/5 = 80 seconds = T3
LCM(T1, T2, T3) = 400 seconds
Time taken by them to meet at starting point for the first time = 400 seconds.

Shortcut 49:
Time taken to meet in a circular race for first time

LCM(RT_1,RT_2,….)

RT1 = Relative time taken for first and second persons to meet.
RT2 = Relative time taken for second and third persons to meet.
Question:
Three persons participate in a race on a circular track of length 400m. They can run at a speed of 2mps, 4mps and 5mps respectively. How long will they take to meet for the first time?
Answer:
RT1 = D/(S1 ~ S2); RT2 = D/(S2 ~ S3)
D = 400m, length of the track.
RT1 = 400/2 = 200 sec ; RT2 = 400/1 = 400 sec
LCM(RT1,RT2) = LCM(200,400) = 400
Time taken for them to meet for the first time = 400s

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