Bus 1 and Bus 2 run between cities A and B.

let d=distance between cities A and B
bus 1 distance to first meeting=50 miles
bus 1 distance from first to second meeting=(d-50)+30=d-20 miles
bus 2 distance to first meeting=d-50 miles
bus 2 distance from first to second meeting=50+(d-30)=d+20 miles
because both buses take the same time for the pre-meeting leg, and
the same time for the inter-meeting leg, and their speeds remain
constant, their ratios between pre-meeting distance and inter-meeting
distance should be the same
therefore, 50/(d-20)=(d-50)/(d+20)
d^2=120d
d=120 miles

hi, chetan2u
thanks for your solution.
it's much more elegant than mine.
gracie

Let the total distance = D

step 1

A ----------------------M1----------- B
50 D-50

step 2

A ----------M2------------------------ B
D-30 30

Bus 1 : total distance traveled = 50+(D-50)+30 = D+30
Bus 2 : total distance traveled = (D-50)+50+(D-30) = 2D-30

OK so what we have now:
- 2 bus traveling with constant speed (let say R1 and R2)
- the total time traveled by both Bus is equal.

---> Ratio of speed= ratio of distance traveled
R1/R2 = (D+30)/(2D-30)
50/(D-50) = (D+30)/(2D-30)
after solving we get :
D^2 - 120D = 0
D(D-120) = 0
D=0 or D=120 bingo.

Here is my solution

Let speed of 1=A & speed 2=B
since time taken to travel by both 1&2 are equal
then as per case 1
50/A=x/B---------------(M)

case 2
distance covered by 1 after covering 50miles= x+30
distance covered by 2 =50+50+x-30

again their time taken is equal
(x+30)/A=(70+x)/B-------------(N)
putting value of A from (M) & solving we get x=70
thus total distance=x+50===70+50=120 miles

Ans B

gracie

Given: Bus 1 and Bus 2 run between cities A and B. Bus 1 leaves city A at the same time Bus 2 leaves city B,
each at a constant rate of speed. Their first meeting is 50 miles from city A. After reaching their
respective destinations and immediately turning around, their second meeting is 30 miles from city B.

Asked: What is the distance in miles between city A and city B?

Let the speed of Bus 1 & Bus 2 be v1 & v2 miles per hour respectively and distance in miles between city A & city B be D miles.

First meeting: - 
City A <---------------50 miles ----->First meeting point-------------------------City B 
Distance travelled by Bus 1 = 50 = v1 * t1
Distance travelled by Bus 2 =D- 50 = v2 * t1
v1/v2 = 50/(D-50)

Second meeting: 
City A------------------------Second meeting point <----------30 miles ----------->City B
Distance travelled by Bus 1 = D+30 = v1 * t2
Distance travelled by Bus 2 = D+(D-30) = v2 * t2
v1/v2 = (D+30)/(2D-30)

50/(D-50) = (D+30)/(2D-30)
(2D-30)50 = (D+30)(D-50) 
100D - 1500 = D^2 -20D - 1500
D^2 -120D =0
D(D-120) = 0
D = 120 miles

IMO B­­

­Why can't we do 50/(D - 50) = 30/(D - 30) instead of adding total distance on trip 2.
I took distances covered in both trips separetely. 
I'm getting confused, pls help. 
 

 

­We are equating the ratio of the distances covered by two buses before the first meeting (X) point to the ratio of the distances covered by two buses before the second meeting point (Y). Thus, it should be:

A------​​​​​​​--​​​​​​​--(50)​​​​​​​--​​​​​​​--​​​​​​​--​​​​​​​--​​​​​​​--X​​​​​​​--​​​​​​​------​​​​​​​--​​​​​​​--​​​​​​​--​​​​​​​--Y​​​​​​​--​​​​​​​--​​​​​​​--(30)​​​​​​​--​​​​​​​--​​​​​​​--B

The above directly gives d = 120.

Answer: B.

­