Proportion of Residents in Each Region

  • Thread starter Nope
  • Start date
In summary, the conversation discusses a problem where a country is divided into three regions and each year a certain proportion of residents move between the regions. The question is about the long-term settlement of residents in each region. Through solving simultaneous linear equations, it is determined that in the long run, approximately 0% of residents settle in the Upper region, 60% in the Lower region, and 40% in the Central region.
  • #1
Nope
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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..
 
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  • #2
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?
 
  • #3
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
 
  • #4
Nope said:
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?
 
  • #5
Dick said:
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
 
  • #6
Nope said:
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.
 
  • #7
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?
 
  • #8
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?
 
  • #9
the final population of Uf is 3/4 of the initial population(Ui)?
 
  • #10
Nope said:
the final population of Uf is 3/4 of the initial population(Ui)?

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.
 
Last edited:
  • #11
if constant,do you mean the answer for U is 3/4?
 
  • #12
Nope said:
if constant,do you mean the answer for U is 3/4?

Does U=3/4 satisfy U=(3/4)U?
 
  • #13
no...
sorry , i still don't get it..
 
  • #14
Nope said:
no...
sorry , i still don't get it..

U=(3/4)U. Solve for U. Use algebra. How would you do that?
 
  • #15
1=3/4?
that's what i am confusing
or do you mean Uf/Ui=3/4? Uf=4 Ui=3?
 
  • #16
oh, I think I got it...
U=0
0=(3/4)*0?
 
  • #17
Nope said:
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.
 
  • #18
ok ,i got 60% for L and 40% for C
But I don't understand why L=L,C=C in the long run...
 
  • #19
Nope said:
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?
 
  • #20
the answer is same,
I see..
what if U is not equal to zero,
like U=3/4U+1/2L
 
  • #21
Nope said:
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
 
  • #22
Thanks, I got it now :)
 

FAQ: Proportion of Residents in Each Region

What is the definition of "Proportion of Residents in Each Region"?

The proportion of residents in each region refers to the relative size or percentage of the population that resides in a particular geographical area or region. It is often used as a measure of population distribution.

How is the proportion of residents in each region calculated?

The proportion of residents in each region is calculated by dividing the number of residents in a specific region by the total population of the area and multiplying by 100. This gives the percentage of residents in that particular region.

3. Why is it important to know the proportion of residents in each region?

Knowing the proportion of residents in each region is important for understanding the distribution of a population and its impact on resources and services in a particular area. It can also provide insight into social and economic patterns within a region.

4. How does the proportion of residents in each region vary across different countries?

The proportion of residents in each region can vary significantly across different countries, depending on factors such as population growth, migration, and government policies. In some countries, there may be a more even distribution of residents across regions, while in others, there may be significant disparities.

5. Can the proportion of residents in each region change over time?

Yes, the proportion of residents in each region can change over time due to various factors such as population growth, migration, and urbanization. Changes in government policies and economic opportunities can also impact the proportion of residents in each region.

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