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#1 2016-09-16 14:25:47

Annar
Guest

Primary school task help needed

Hey guys,

I am no maths genius and need help with this itsy bitsy problem. I know I can easily find the solution by adding them one by one but I was wondering if someone can come up with a quicker solution to this problem, some sort of a formula I guess.

This is the problem:

A train will hold 78 passengers. If it starts out empty and picks up one passenger at the first stop, two passengers at the second stop, 3 passengers at the third stop and so on. After how many stops will it be full?

Thanks, appreciate the help

#2 2016-09-16 15:07:48

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Primary school task help needed

Hi;

The sum of the first n numbers written in summation notation is

So we have a formula. We take and set it to 78.

Multiply both sides by 2.

Expand out.

We have a quadratic equation and we solve using factoring, completing the square or the quadratic formula.

We get n = 12 and n = - 13

We of course choose the positive answer, so n = 12. That means

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 78

The train is full at the 12th stop.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2016-09-16 20:23:45

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: Primary school task help needed

hi Annar

Welcome to the forum.

Let's say there are n stops.  So the summation becomes 1 + 2 + 3 + ... + n      or       n + (n-1) + ... + 3 + 2 + 1

If you add these two sums together, first terms together, then second terms together, and so on,  you'll get (1+n) + (2 + n-1) + ...  + (n+1).  Every bracket comes to the same thing (n+1) and there are n of them.  So double the total comes to n(n+1).  So we have this formula

To solve this,  double to get n(n+1) = 78 x 2 = 156.  So you want two consecutive numbers that multiply to 156.  Only one positive answer to this: n=12, as 12x13=156.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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