Maths SOS

[Casparina]

We're struggling with a maths question in our household. It doesn't look as if it should be that hard! Can anyone help?

The trains did 140m in 14seconds.
140m in 14secs
600m in 60secs (1minute)
(times by 60 to make an hour)
36000m in 1 hour
(Divide by 1000 to turn metres in km)
36km in 1 hour
so 36kmph

(not an expert, might be completely wrong but this was my initial thoughts)

Ah I had a different answer @RunningFromInsanity It is 18 kph I think

If they pass at the same speed they are each doing 70m in 14 seconds - which factors up to 7k in 23.3333 minutes or 21 k in 70 mins - divide both by 7 and multiply by 6 give 18k in 60 mins. If you had one train passing a static object that fast it would be going twice the speed, hence the inclusion of 36kph as an option I think.

I'm not sure my old maths teacher, Miss Harrison, would be over-impressed wit my method tho!

By 'cross' does that mean the time for them to completely pass? In which case I think clary is out by a factor of 2 and it's 36kph?

@RunningFromInsanity has the right answer

They are both going at X m/s

The relative speed is therefore 2 x Xm/s

2x = (140 + 140)/14 (metres over seconds)
2x = 280/14
2x = 20
x = 10 m/s

10m/s is 36kmph.

It’s either 18 or 36 depending on whether the distance is 70 or 140

Well but my thought was, if the train travelled for 14 seconds past a static point, that would be one thing, but if that point were also moving in the opposite direction, it would take less time to pass? Happy to accept that I am wrong tho.

Better: If I am running at 10kph, and I run past a 1km wall (not moving) then it will take me 6 minutes; if I run at the same speed past a 1km long column of runners, also running at 10kph in the opposite direction, that will sure take less time?

The distance covered is 140 x 2, because both trains are travelling at the same speed and will cross in 14 seconds. They are currently both 140m away from the crossing point. The crossing point being the point at which they first reach the same point.

My reasoning was that yes it would be quicker to pass a moving object but the distance travelled would still be the same?

But I have no real conviction in that theory, wouldn’t be surprised if it’s wrong.

I think I agree with those saying 36km/h.

@clary They aren't crossing a point, they are crossing something 140m long.
(So if one was stationary the other train would have to travel 140m to get level and then a further 140m to clear. As they are both moving they each have to travel 140m.)

Thanks so much to you all. This is exactly the debate we have been having around our kitchen table but got in a muddle with what to halve and what to double etc. @RunningFromInsanity 's relative speed explanation has really helped us make sense of it.

Can I ask just one more that we're stuck with, if people don't mind? With this one we know that the correct answer is C but we thought it was D (5 hours rounded from 5.08 hours) having converted the filling rates to the lowest common multiple.

Absolutely no idea with that one, I’m afraid! 😂

Lowest common multiple is 12 days

Hose one fills the pool 48 times
Hose 2 - 6 times
Hose 3 - 4 times
Hose 4 - 3 times

Total 61 times in 12 days, or 288 hours

288/61 = 4.72

Really neat solution @PatriciaHolm

In an hour each hose will fill the following fractions of the pool:

1/6
1/48
1/72
1/96

So altogether in a hour, you can add those fractions together to work out how much of the pool is filled. To add them, they can all be converted to x/288 so:

48/288 + 6/288 + 4/288 + 3/288 = 61/288

So in an hour, 61/288 worth of the pool will be full. The pool will be completely full in 288/61 hours which is 4.72 hours. So 4 hrs and 45 mins to the nearest quarter hour.

I did the last one slightly differently, but same answer:

In one hour, hose 1 fills 1/6 of pool
In one hour, hose 2 fills 1/48 of pool
In one hour, hose 3 fills 1/72 of pool
In one hour, hose 4 fills 1/96 of pool

common denominator is 288 hours

48/288 + 6/288 + 4/288 + 3/288 = 61/288 of pool filled in an hour.

So to find out how many hours, it must be reciprocal of that 288/61 (i.e., how many 61s needed to fill up the 288).

61x4= 244, so 4 hours with 44/61 left. 44/61 is approx 45/60, so close enough to 45 min.

Brilliant. Thank you all so much. You clever people make it look so simple!

Ah that makes sense. In my running example I am just a person not a big line of people! OK 36kph well done everyone

Not to put a spanner in the works but I put the question to my family and it’s a 2/3 split in favour of 18kmph.

My maths might be right but my physics might not be.

Arguments include-
Closing speed
The 36kmph is for both trains, need to half for each train
There was a rogue shout for the distance being 280m as that’s the length of both trains combined.

Where did you find these questions? I would like to have a go at some more!

There was a rogue shout for the distance being 280m as that’s the length of both trains combined.

That's correct if by 'cross' you mean 'totally pass'.

@RunningFromInsanity

Imagine one train being still.
The other will have to travel 280m to get all the way past.
As they are both moving then each train has to travel 140m.

Use 2 remote controls to demonstrate it to yourselves.

The distance is 280m, it's not a "rogue shout". "Cross" means reach a point where the two trains are on the same point on the track at the same time. Essentially the point at which they would collide if they were on the same track. The question says that will happen in 14 seconds and both trains are 140m long. So currently they are one trains length each away from that point, so 2 x 140m apart, 280m.

The time taken by two running trains in crossing each other is calculated by dividing the sum of the lengths of two trains by the total speed of the two trains.

The equation is (L1+L2)/(S1+S2) = T

In this case we know that is 280/x = 14

280/14 = 20, which in this case is metres per second, so each train is running at 10mps (20/2).

Which is 36kph.

So I was correct in my original answer of 36kmph?

Yes!