The Two Trains and How Fast

Agnishom · 2012-07-12 01:16:21

Here's my approach to it but I can't find what's wrong:

Let time taken by the first train TO COMPLETE THE WHOLE JOURNEY be t1
Let time taken by the second train TO COMPLETE THE WHOLE JOURNEY be t2
Let the distance between London and Liverpool be d
Let the time when they had met each other be f

Now:

Because at the moment of passing each other the distance both trains have covered add up to d
Because Train1 takes one hour more after the passing each other moment
Because Train2 takes four hours more after the passing each other moment

I shall call the above Equation 1, 2 & 3 respectively

From Equation 2:

or
or

From Equation 3:

or
or

Therefore, we can say that:

Putting t2 = (t1 + 3) and f = (t1 - 1) in Equation 1:

After fooling around with it you get:

Now according to the brute force method of Quadratic equations, I get:
and

Ultimately we get the following things:

What are the mistakes(if any), I have commited in my above calculations?

Now the most important thing:
The puzzle told us to find how much faster is one train running than the other
and the answer given in the book is this:

How can I come to the conclusion in the answer?

bobbym · 2012-07-12 01:55:00

Hi Agnishom;

Bob · 2012-07-12 06:23:10

hi Agnishom

All the way to the quadratic is correct.

But 'd' is a common factor and is not = zero so you can cancel it out leaving

so t1 = 3 not 3d. (d is a distance so how could the time be equal to 3 x it ?)

implies t2 = 6

So the slow train is taking twice as long => twice as fast.

As for your method, I think you have over complicated it.

(i) I always like to have a distance / time graph as it helps me to see what unknowns I have

(ii) Don't introduce more unknowns than needed as it just makes the algebra worse.

So I had time to crossover = t, distances covered to this point c and d.

Then

and

Divide one equation by the other and both c and d are eliminated giving

So the times for the whole journey are 2 + 4 and 2 + 1

Bob

Agnishom · 2012-07-12 23:56:48

Hi Bob,
How do you know that

Please explain the above
and
You Asked:

(d is a distance so how could the time be equal to 3 x it ?)

My Answer
I am getting this because I have not considered the units

anonimnystefy · 2012-07-13 00:38:14

Hi Bob

I think that after getting a relation between t and d he can treat them as unitless variables.

Bob · 2012-07-13 08:44:21

hi Agnishom and Stefy,

Yes, I understand your argument about units. But look again at the quadratic

so

so either d = 0 or t1 +1 = 0 or t1 -3 = 0

As we know d isn't zero and the negative doesn't fit the problem we can conclude t1 = 3.

The 'd's have gone from the problem.

How did I get d/t = c/1 ?

That's distance / time = speed for each part of the journey (speed is the same for both parts)

Bob

anonimnystefy · 2012-07-13 08:46:25

Yes, he didn't solve it properly. I think he put only 2 instead of 2a in the quadratic formula. A common error.

Bob · 2012-07-13 09:06:28

Maybe.

I keep a look out for common factors like 'd' and cancel them out if I can (ie. if the common factor is not zero)

This quadratic factorises without the formula. In my youth there were no calculators so using the formula meant looking up the square root in a table and then a 'nasty' calculation with decimal places. So I always try to factorise if I can.

Bob

anonimnystefy · 2012-07-13 09:10:15

But if the expression can be factorized then the quadratic formula won't have any nasty decimals.

Bob · 2012-07-13 09:19:46

Arrhhh, you spotted the flaw in my argument.

But if I factorise it first then I haven't even had to reach out for my table book. It's just a habit that is ingrained into me. Like using my knowledge of tables rather than using a calculator. It's just an old fashioned way of working, but some of us 'olds' think it is better. Don't you get told that sometimes?

Bob

anonimnystefy · 2012-07-13 09:23:50

Hi Bob

Well, what if you get something nasty. You can always do the formula, then, if you get a nasty root, you just pull out the tables. It is the most productive way, because an expression might not be factorable and if is, the factors may not always be obvious.

Bob · 2012-07-13 09:37:10

You do it your way and I'll do it mine.

Either way there's no 'd'.

Bob

anonimnystefy · 2012-07-13 09:43:29

That we agree on.

Agnishom · 2012-07-13 17:07:36

anonimnystefy wrote:

Yes, he didn't solve it properly. I think he put only 2 instead of 2a in the quadratic formula. A common error.

Yes thats true
That was what I was eager to know about

bobbym · 2012-07-13 22:45:47

Hi Agnishom;

Happens to everybody.

Agnishom · 2012-07-21 16:36:29

Oho! Really?

anonimnystefy · 2012-07-21 16:37:41

Of course. I must've forgotten the 'a' at least 40 times.

bobbym · 2012-07-21 18:13:32

40 times or 10^40 times?

anonimnystefy · 2012-07-22 02:15:07

At least 40 times. 10^40 is too much, though. I give an upper bound of 1000.

Agnishom · 2012-07-26 01:02:05

40 times out of a total of how many?

bobbym · 2012-07-26 01:50:42

Hi;

anonimnystefy either has a very dry sense of humor or he missed the point completely by overlooking the reference to the anonimnystefy constant.

Agnishom · 2012-07-26 02:24:22

What is "anonimnystefy constant"?

bobbym · 2012-07-26 05:43:37

It has a numerical value of 10^40. Other than that I do not know much about it. It first showed up in a thread where you were given 20 guesses to guess someone's occupation and identity. He needed or took more guesses, hence the anonimnystefy constant.

anonimnystefy · 2012-07-26 06:00:21

bobbym is the one to propose that the number of guesses for we should be 10^40. But I grabbed the name for it before he could. But, I recently coined the term bobbym constant for the number 123456787654321.

bobbym · 2012-07-26 06:03:24

I already have a constant named after me. There is a rule in math. One constant per person. There are no exceptions.

Math Is Fun Forum

#1 2012-07-12 01:16:21

The Two Trains and How Fast

#2 2012-07-12 01:55:00

Re: The Two Trains and How Fast

#3 2012-07-12 06:23:10

Re: The Two Trains and How Fast

#4 2012-07-12 23:56:48

Re: The Two Trains and How Fast

#5 2012-07-13 00:38:14

Re: The Two Trains and How Fast

#6 2012-07-13 08:44:21

Re: The Two Trains and How Fast

#7 2012-07-13 08:46:25

Re: The Two Trains and How Fast

#8 2012-07-13 09:06:28

Re: The Two Trains and How Fast

#9 2012-07-13 09:10:15

Re: The Two Trains and How Fast

#10 2012-07-13 09:19:46

Re: The Two Trains and How Fast

#11 2012-07-13 09:23:50

Re: The Two Trains and How Fast

#12 2012-07-13 09:37:10

Re: The Two Trains and How Fast

#13 2012-07-13 09:43:29

Re: The Two Trains and How Fast

#14 2012-07-13 17:07:36

Re: The Two Trains and How Fast

#15 2012-07-13 22:45:47

Re: The Two Trains and How Fast

#16 2012-07-21 16:36:29

Re: The Two Trains and How Fast

#17 2012-07-21 16:37:41

Re: The Two Trains and How Fast

#18 2012-07-21 18:13:32

Re: The Two Trains and How Fast

#19 2012-07-22 02:15:07

Re: The Two Trains and How Fast

#20 2012-07-26 01:02:05

Re: The Two Trains and How Fast

#21 2012-07-26 01:50:42

Re: The Two Trains and How Fast

#22 2012-07-26 02:24:22

Re: The Two Trains and How Fast

#23 2012-07-26 05:43:37

Re: The Two Trains and How Fast

#24 2012-07-26 06:00:21

Re: The Two Trains and How Fast

#25 2012-07-26 06:03:24

Re: The Two Trains and How Fast

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