**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