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?
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?
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.
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?
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?