How much faster is Train 1 than Train 2?

Question

Two trains, on opposite sides of parallel tracks, leave their stations at the same time and are headed towards each opposing station. Train 1 is faster than Train 2, and when they pass each other on the tracks Train 1 has one hour to reach Station 2, while Train 2 has four hours to reach Station 1. How much faster is Train 1 than Train 2?I‘ve found the answer (through brute force) but can’t seem to mathematically represent it.

Answer 1

 

t is the time at which they meet.

R is the speed of the faster train

r is the speed of the slower train

 

The quantity we are trying to find is R-r, the difference in their speeds.

 

They will complete the distance to the 

other station in 1 and 4 hours respectively.

Which means their traveling times are 1+t and 4+t respectively.

 

So the distance between the train stations is:

 

(4+t)r = (1+t)R = (R+r)t <--- BOTH trains cover the same distance in the same time at their combined speeds

 

 

 

Solving equations #1 and #2

 

(4+t)r = (1+t)R

4r + rt = R + Rt

4r - R = Rt - rt

4r - R = t(R-r)

 

(4r-R)/t = R-r

 

 

Solving equations #1 and #3

 

(4+t)r = (R+r)t

4r + rt = Rt + rt

 

4r = Rt

 

 

 

Solving equations #2 and 3

(1+t)R = (R+r)t

R+Rt = Rt + rt

R = rt

 

 

The three resulting equations are:

(4r-R)/t = R-r

4r = Rt 

R = rt

 

Plugging the third of these into the 1st:

(4r - rt)/t = rt - r

r(4-t)/t = r(t-1)

(4-t)/t = t-1

 

4-t = (t-1)t

0 = t^2 - t - 4 + t

0 = t^2 - 4

0 = (t+2)(t-2)

 

t+2 results in negative times

t=2

 

 

Plugging this result into previous equations

show that the first train is TWICE as fast as the second train.

 

Answer 2

good question.

 

Let speed of train1 is "a" and train2 is "b".  the time till they met was "y".

 

         Speed * time = distance

trian1   a *y = ay miles

train2   b* y = by miles

 

train1 has distance of "by" to go.

train2 has distance of "ay" to go.

 

Now

       Speed * time = distance
trian1   a * 1 = by miles  therefore, y=a/b.
train2   b * 4  = ay miles therefore, y=4b/a

 

now,  y = a/b=4b/a

so a^2 = 4b^2

that means a = sqrt(4)*b=2b

 

a =2b