Puzzle | Three Brothers Travelled by Road

Question :
Three brothers, R, S and M, travelled by road. They all left the college at the same time – 12 noon. The description of the motions of the three are detailed below –

Name	R	S	M
Phase I	Bus for 2 hours @ 10 mph	Bike for 1 hour @ 30 mph	Foot for 3 hours @ 3.33 mph
Phase II	Bike for 1.5 hours @ 40 mph	Foot for 3 hours @ 3.33 mph	Bus for 3 hours @ 10 mph
Phase III	Foot for 3 hours @ 3.33 mph	Bus for 4 hours @ 10 mph	Bike for 2 hours @ 30 mph

Using the information answer the two questions –

1. When did R overtake S?
2. If R travelled by bike instead of the foot in the last leg of his journey (for the same distance as he had covered by foot), what is the difference in the times of R and M to cover 90 miles?

Explanation :
Formula to find the distance is :

Distance = Speed * Time

By using this relation we can find the distance covered at the end of each hour.

Example –
Distance covered by R at the end of the first hour :

Speed * Time = 10 * 1 = 10 miles

Distance covered by R at the end of the second hour :
Distance covered till the first hour + Distance covered during the second hour

= 10 + 10 * 1 = 20 miles

Distance covered by R at the end of the third hour :
Distance covered till the second hour + Distance covered during the third hour

= 20 + 40 * 1 = 60 miles

Distance covered by R at the end of the fourth hour :
Distance covered till the third hour + Distance covered during the fourth hour

= 60 + ( 40 * 0.5 + 0.5 * 3.3 ) = 81.66 miles

During the fourth hour, half of the time is travelled at the speed of 40mph and another half of the time for 3.3mph

Distance covered by R at the end of the fifth hour :
Distance covered till the fourth hour + Distance covered during the fifth hour

= 81.66 + 3.3 * 1 = 85 miles

Distance covered by R at the end of the sixth hour :
Distance covered till the fifth hour + Distance covered during the sixth hour

= 85 + 3.3*1 = 88.33 miles

Distance covered by R at the end of the seventh hour :
Distance covered till the sixth hour + Distance covered during the seventh hour

= 88.33 + 3.3 * 0.5 = 90 miles

During the seventh hour, half of the time is only travelled because the total time taken to reach the destination is 6.5hrs for R.

In the same distances for S and M at the end of each hour is calculated.

The table below shows the distance covered by each person at the end of each hour.

Hour	R	S	M
1	10	30	3.33
2	20	33.33	6.66
3	60	36.66	10
4	81.66	40	20
5	85	50	30
6	88.33	60	40
7	90(6.5 hrs)	70	70
8		80(8 hrs)	100(8 hrs)

1. It is evident that R would overtake S between 2 and 3. At x, S is ahead by 13.33 km.

   Relative speed between 2 – 3 
   = 36.66 kmph.
   
   The time required 
   = 13.33/36.66 of the hour
   = 4/11 of the hour 
   = 22 mins 

   The time at which R overtakes S

   = 12 + ( 2 + 4 / 11 ) 
   = 2 : 22. (approx)

2. R would cover 90 miles in 3 hours 45 minutes. (Travels at a speed of 10mph for 2hrs and at a speed of 40mph for 1.45hrs to travel 90miles)

   M would cover 90 miles in 7 hours 40 minutes. (Travels at a speed of 3.3mph for 3hrs, at a speed of 10mph for 3hrs and at a speed of 30mph for 1.40 hrs to travel 90 miles)

   Time difference = 3 hours 55 minutes.