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Mathegocart

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Registered: 2012-04-29

Posts: 2,226

Track question

1. A circular track is 1000 yards in circumference. Cyclists A B and C start at the same place and time at the following rates per minute, respectively, 700 YARDS, 800 yards, and 900 yards. What is the LEAST number of minutes it takes for all 3 to be back again?

Last edited by Mathegocart (2016-07-10 13:12:47)

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Fruityloop

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Registered: 2009-05-18

Posts: 143

Re: Track question

10 minutes.

Bob

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Registered: 2010-06-20

Posts: 10,626

Re: Track question

hi Mathegocart

Let's say they meet after T mins at a point x around the track.  Also A has completed n complete laps (+ x); B m laps; and C p laps.  Then

Eliminate x by subtractions and we get

Clearly p > m > n (and all are whole numbers) so inspection  shows p = 3; m = 2; n = 1 will work.

Bob

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Children are not defined by school ...........The Fonz
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thickhead

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Registered: 2016-04-16

Posts: 1,086

Re: Track question

x should be zero because they have to meet at the starting point.. That is taken care of by T which need not be an integer.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
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Bob

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Registered: 2010-06-20

Posts: 10,626

Re: Track question

Oh.  I see what you mean.  I interpreted 'back again' as meaning 'back together' rather than 'back at the start'.  But, as I've eliminated x anyway, it doesn't affect my answer. 

Bob

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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

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

only for p=9, m=8 and n=7 ( the least values) for which T will be same for all.T=10 minutes.

Last edited by thickhead (2016-07-10 21:05:25)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Bob

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Registered: 2010-06-20

Posts: 10,626

Re: Track question

hi thickhead,

Thanks.  I get it now.  I am trying to find another (not minimal) solution, just to check my understanding.  Is 18,16,14 OK for this?

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

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

Meeting for the second time.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Track question

The problem is best done using congruences, we must solve

The answer is 10.

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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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

I would prefer to say the time spent must be L.C.M. of time taken by each for one complete round i.e 1000/900,1000/800 and 1000/700 Their proper fractions are 10/9, 5/4 and 10/7. For getting L.C.M. we have to ignore denominators and take L.C.M. of numerator which is 10.
correction Not to ignore denominator. we have to take L.C.M. of numerator/ G.C.D. of denominator.

Last edited by thickhead (2016-07-12 00:14:06)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Track question

if that is what you like then go ahead but can anything beat

Subtract congruence 2 from congruence 1 and we get:

Now what number must x be so that 100 x is divisible by 1000? Obviously 10 and we are done.

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

hi bobbym,
That is correct but one has to keep in mind that x need not be an integer, it can be a fraction.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

If the speeds are 360,540 and 720 yards/minute  How do we go ahead with the following?

Last edited by thickhead (2016-07-13 20:44:51)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Track question

Hi;

I am getting x = 50 for that one.

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

But they will meet at the starting point after they complete 2,3 and 4 rounds respectively  after a lapse of 50/9 minutes. Your 50 is correct but we have to divide by 9. How do we get this number 9? It is not GCD also. It may be the number obtained after 180 is divided by GCD of 180 and 1000 or by  180*50/1000. I am not much conversant with use of mod calculation.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Track question

720 mod 1000 is not 0

The original problem is in terms of distance (1000). You want to think about it in terms of degrees. The system of congruences given in post #13 is not correct for that.

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

I have replaced 900,800,700 by 360,540,720 a different set of speeds.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Track question

Then 2 seconds is 720 ft not 1000ft. They are not all at the starting point together.

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

After 50/9 minutes they cover distance of 360*50/9=2000, 540*50/9=3000 and 720*50/9=4000 yards respectively.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Track question

Mods are like remainders, when we say what is 13 / 7 we can say 1.857142857142857... or 1 with a remainder of 6. Modular arithmetic are always integers.

What you posted in #13 should be answered in whole numbers, that is why the answer is 50.

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Track question

The GCD might need some work too... What if the speeds are not whole numbers?

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

If they are not whole numbers, convert them to proper fraction and then GCD of numerator/LCM of denominator should be used.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Track question

Hi;

Did you try that on a problem yet?

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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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Track question

Last edited by thickhead (2016-07-16 16:13:37)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}