On Friday I posted this puzzle….

New York and Boston are 220 miles apart. A train leaves Boston for New York and travels at 65 mph. One hour later, another train leaves New York for Boston and travels at 55 mph. Assume the tracks are straight paths and the trains maintain a constant speed.  How far apart are the trains 1 hour before they meet?

If you have not tried to solve it, have a go now.  For everyone else, the answer is after the break…

Think about it backwards. An hour before they meet, one train is 65 miles away from the meeting point, while the other is 55 miles. Add the two distances together and you’ll get 120 miles.  Any other ways of working out the answer, or other solutions?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for theKindle (UK here and USA here) and on the iBookstore (UK here in the USAhere). You can try 101 of the puzzles for free here.

I reckon that’s the only way. The only tricky part is not getting confused thinking you have to work out where on the track they are going to meet…

2. David Gillion says:

Situation is same as if one train at rest and other approaching it at 120mph.

3. Goliath says:

I started with the usual – scribbling, doodling, yelling, until I realised it was really that easy. Yay!

4. Anonymous says:

They meet after 2.3 hours. Regarding where they were after 1.3 hours, I calculated a distance about 119 miles.

T1 * 65 + T2 *55 = 220 ; T2=T1-1 –> Meeting Point 2.3 hours
D= 220 – 1,3* 65 – 0,3*55 =119 miles

They I read the comments and realised there must be an easier way. Got it after reading again. Good puzzle!

5. edgar1975 says:

Yep, that’s how I got it.

6. Charlie says:

In addition to working out the answer this wya, you should really prove that neither train is still at a station one hour before they meet. Granted it is very easy to do, but nonetheless you have to be certain otherwise the answer to the puzzle would be different.

1. Chris says:

I thought that was going to be the trick, until I did the maths.

For those of you that have found Richard’s “puzzles” a little pedestrian of late, I understand that the Daily Star has a crackin’
Sudoku.
And you always get the answer the next day.
And the answer is always right.
Those wordsearches are pretty good too.

1. Anonymous says:

Haha so true! 🙂

8. Anders says:

I summed the infinite series!

Oh, wait, that doesn’t work here. Never mind…

1. RModule says:

I laughed.

9. Mickey D says:

When you suddenly realise it’s that easy it’s great!

10. JimC says:

But do we know which train always tells the truth and which train always lies?

11. Lazy T says:

I converted the speeds and timings into morse code then into roman numerals, divided by the rate of tree growth and got the answer… £4.67.

12. Shaun says:

That’s not the answer I get, unless I’ve misunderstood the question.

As the first train leaves Boston earlier and travels faster than the second one, the second train will only be able to meet the first one at the station in New York 220 miles away.

It is travelling at 55mph, it will be 55 miles away from the first train one hour before they meet (assuming it’s still sitting stationary at the station).

1. arcur says:

You missed that they are traveling in opposite directions.

2. Shaun says:

So I did!

13. I shifted the reference frame so that one was stationary, t’uther at 120mph.

14. Luis says:

Assume the trains are together at [time = 0]. Train A heads to Boston at 65mph, and train B heads to NYC at 55mph. How far apart are they going to be at [time = 1 hour]?

15. Richard says:

I did it basically the same way but from a different perspective: I summed their speeds to get a “closing speed” of 120mph. At that point it’s obvious again to say that one hour before meeting they must be 120 miles apart.

16. ivanolier says:

Train A: Xa=65t
Train B: Xb=200-55t

Then, solving: Xa = Xb -> time when Xa=Xb -> 65t=200-55t, then To = 5/3,

now, 1 hour before, T1 = 5/3 – 1 = 2/3.

Consequently, Xa1 = 65(2/3) = 130/3, and Xb1 = 200-55(2/3) = 490/3.

Finally: Xb1 – Xa1 = 490/3 – 130/3 = 120miles 🙂

17. Beaky says:

Gee, it seemed simple to me – closing speed is 120 mph,
ergo 1 hr earlier they were 120 mi apart. That took 3 sec.
But then I had to make sure that train B had left the station an hour before they met. That was a bit harder – 10 mins.

18. One Eyed Jack says:

Politician A begins campaigning in Chicago on February 2nd. Politician B begins campaigning in Boston on February 3rd.

They meet in New York for a debate on February 15th.

Calculate the total amount of Bull spewed by each candidate between the start of the campaigns and the start of the debate.

It is, of course, a trick question. Modern science has never been able to calculate a politician’s capacity for Bull.

19. Beaky, I don’t know what you spent ten minutes on to prove that that train B had left the station but I just asked myself how far apart they were when train B left the station.

After Train A has been travelling an hour at 65 mph it’s 220-65=155 miles from Train B as Train B gets going. Since that’s greater than the 120 miles we’re golden.

30 Seconds.

Frankly it was a disappointment that this twist wasn’t included to catch out those with the slightly more obvious answer.

e.g.

Train leaves Cardiff on a 150 mile journey to London at 80 mph. An hour later a Train Leaves London travelling in the opposite direction at 60mph. What distance are they apart, an hour before they meet. Obvious answer 140 miles, Correct answer…

1. John Loony says:

110 miles (they meet 30 miles from London) … I think

20. SteveG says:

I don’t even want to tell you how I went about getting the correct answer. But I will.

I looked at the original distance when New York (n) set out=155. Then I calculated how far each would go until they met: Since each would take the same amount of time I used T=D/S. knowing each one’s speed and that their times are equal, I came up with D(n)/55 = D(b)/65. Then, knowing that D(n) + D(b)= 155, I substituted and came up with D(n)/55=(155-D(n))/65. After calculating each distance, I calculated each time (about 1.29 hours) then I subtracted an hour (0.29 hours) and multiplied it times each speed to find the distance each one traveled. I added them together and subtracted it from 155 and got 120 miles! (now that’s taking the scenic route…)

“Think about it backwards” yeah, that was a good idea.

I see from the comments I wasn’t the only one. Well at least that’s something…

21. Physics Teacher says:

This would be a far more interesting question if you chose two cities that were closer than 120 miles, so that one train was still in the station an hour before they met.

110 miles out of Swansea (The graveyard of ambition)?

23. Mike says:

All you have to do is calculate the closing speed in miles/hour. Which is as simple as speed (train 1) plus speed (train 2).

24. John Loony says:

Hahaha at all the dilly twogliaks who worked it out the long way! Don’t you know Richard by now?

25. Trick question. The two trains are on separate tracks, so they’ll never meet.

26. Tim says:

Hi Richard,

I live in Hong Kong and thought to might like to see this:
http://www.squarefoot.com.hk/haunted/
A lot of the locals check their flats-to-be for hauntings etc!

On a more cheeky note, can you recommend a uni that does a MSC or PHD in Parapsychology or similar by distance?

Cheers,

Tim