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

Question

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

A) 90
B) 120
C) 125
D) 150
E) 180

(source:self)

chetan2u · Accepted Answer

Hi,

let the distance be x between A and B...  
so Bus 1 travels the entire distance x to B and then turns back and travels 30 miles before both busses meet again..
  so Bus 1 travels x+30 miles...(i)...

Also..
when they meet first time, both combine travel x miles, out of which 1 travels 50..
then they reach the end cities and travel back, so when they meet second time, they have travelled 3x distance..
If bus 1 travelled 50 in x, it will travel 3*50 in 3x...
  So bus 1 travels 150 miles...(ii)..

from (i) and (ii)
150 = x+30..
x=120..
B

KarishmaB · Answer

You can simply use ratios to solve it.

When talking about same time, ratio of distance covered will be same as ratio of speeds (which are constant). 
So let's say distance between the first meet point and B is x.
Speed of Bus1/Speed of Bus 2 = Distance covered by Bus 1/Distance covered by bus 2 =  \frac{50}{x} = \frac{(x + 30)}{(50 + x + 50 - 30)} 
50(x + 70) = x(x + 30)
On solving, you get x = 70
Distance between A and B = 50 + x = 50 + 70 = 120. 
Answer (B)

gracie · Answer

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

gracie · Answer

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

Abkhazian · Answer

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.

rohit8865 · Answer

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

Kinshook · Answer

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

Sahith_Manikanta · Answer

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.

Bunuel · Answer

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 
 
 \frac{50}{d - 50} = \frac{(d - 50) + 30}{50 + (d - 30)}  
 
The above directly gives d = 120.

Answer: B.

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

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