Proportion of Residents in Each Region

[Nope]

Homework Statement

Suppose that a country is divided into three regions: Upper, Lower and Central. Each year, one-quarter of the residents of the Upper region move to the Lower region and the remaining residents stay in the Upper region. One-half of the residents of the Lower region move to the Central region and the remaining residents remain in the Lower region. Three-quarters of the residents of the Central region move to the Lower region, and the remaining residents stay in the Central region.
In the long run, what proportion of the residents settle in each region?
of the total residents settle in the Upper region,
of the total residents settle in the Lower region, and
of the total residents settle in the Central region.

Homework Equations

The Attempt at a Solution

I have
U=3/4U
L=1/4 U+1/2L+3/4C
C=0 +1/2L+1/4C
I don't know what to do next ,please help..

 

[Dick]

Express that linear system in matrix form. (U,L,C)=M(U,L,C) where M is a 3x3 matrix. What's M? If there is a steady state then M has a eigenvalue of 1. What's the corresponding eigenvector?

 

[Nope]

M is
3/4 0 0
1/4 1/2 3/4
0 1/2 1/4
What do u mean "steady state" and "eigenvalue"? I don't think I learn these yet...
btw, is there a simple way to do it?
ty

 

[Dick]

M is
3/4 0 0
1/4 1/2 3/4
0 1/2 1/4
What do u mean "steady state" and "eigenvalue"? I don't think I learn these yet...
btw, is there a simple way to do it?
ty

Yes, actually, there is. Just solve the equations you have for U, L and C. What does the first equation tell you about U?

 

[Nope]

Yes, actually, there is. Just solve the equations you have for U, L and C. What does the first equation tell you about U?

what about the total population, do i assume 1? or 3?
If 3,
then i got 25%=U , 50%= L, 25%=C
but I don't know how to determine the long run, but I think U is going to be 0

 

[Dick]

what about the total population, do i assume 1? or 3?
If 3,
then i got 25%=U , 50%= L, 25%=C
but I don't know how to determine the long run, but I think U is going to be 0

I hope you concluded U=0 from one of your equations. You don't have to assume anything for the total population. Call it P. So U+L+C=P. Or put P=1 if you just want to work with percentages.

 

[Nope]

no, I just guessing, cause i tried to cube the matrix and multiply it by 1, so in the third year ,U is decreasing, L and C is increasing
but how do i get the long run?

 

[Dick]

If you don't know 'eigenvalue' and 'eigenvector' forget about the matrix. Just solve the original equations you wrote down.

U=3/4U
L=1/4 U+1/2L+3/4C
C=0 +1/2L+1/4C

As I said before, what does the first equation tell you about U?

 

[Nope]

the final population of U_f is 3/4 of the initial population(U_i)?

 

[Dick]

the final population of U_f is 3/4 of the initial population(U_i)?

That's true if you mean U_i is the population at the beginning of the year and U_f is the population at the end of the year. The problem says "In the long run". They are implying that the population of each area will settle down to a constant value. So that equation becomes U=(3/4)U.

 

[Nope]

if constant,do you mean the answer for U is 3/4?

 

[Dick]

if constant,do you mean the answer for U is 3/4?

Does U=3/4 satisfy U=(3/4)U?

 

[Nope]

no...
sorry , i still don't get it..

 

[Dick]

no...
sorry , i still don't get it..

U=(3/4)U. Solve for U. Use algebra. How would you do that?

 

[Nope]

1=3/4?
that's what i am confusing
or do you mean Uf/Ui=3/4? Uf=4 Ui=3?

 

[Nope]

oh, I think I got it...
U=0
0=(3/4)*0?

 

[Dick]

oh, I think I got it...
U=0
0=(3/4)*0?

That's it. U=0. Now put that into the other equations and find a relation between L and C.

 

[Nope]

ok ,i got 60% for L and 40% for C
But I don't understand why L=L,C=C in the long run...

 

[Dick]

ok ,i got 60% for L and 40% for C
But I don't understand why L=L,C=C in the long run...

Suppose U=0, L=60 and C=40 in one year. Following the instruction in the problem what are U, L and C in the next year?

 

[Nope]

the answer is same,
I see..
what if U is not equal to zero,
like U=3/4U+1/2L

 

[Dick]

the answer is same,
I see..
what if U is not equal to zero,
like U=3/4U+1/2L

This example was particularly easy. In general the subject is called 'solving simultaneous linear equations'. You can probably find something on the web if it isn't covered in you book. Here's a simple one. http://www.themathpage.com/alg/simultaneous-equations.htm

 

[Nope]

Thanks, I got it now :)