Question 1. Population migration
people move between an urban and rural area. In year one, 300 people live in the urban
and 500 people live in the rural area.
very year, (a) half of the people who live in the urban area move to the rural area (the otl
half remain in the urban area);
b) a quarter of the people who live in the rural area move to the urban area (the other th
quarters remain in the rural area).
What is the population of each area for any year ?

Ask by Kirk Howell. in Ghana

Mar 28,2025

Question

Question 1. Population migration
people move between an urban and rural area. In year one, 300 people live in the urban
and 500 people live in the rural area.
very year, (a) half of the people who live in the urban area move to the rural area (the otl
half remain in the urban area);
b) a quarter of the people who live in the rural area move to the urban area (the other th
quarters remain in the rural area).
What is the population of each area for any year ?

Ask by Kirk Howell. in Ghana

Mar 28,2025

UpStudy AI · Accepted Answer

To solve this problem, we can use a system of linear equations to represent the population migration between the urban and rural areas.

Let:
- $$ U_n $$ be the population in the urban area after $$ n $$ years.
- $$ R_n $$ be the population in the rural area after $$ n $$ years.

Given:
- In year 1, 300 people live in the urban area and 500 people live in the rural area.
- Every year, half of the people who live in the urban area move to the rural area, and the other half remain in the urban area.
- Every year, a quarter of the people who live in the rural area move to the urban area, and the other three quarters remain in the rural area.

We can set up the following system of linear equations based on the given information:

1. $$ U_1 = 300 $$ (initial population in the urban area)
2. $$ R_1 = 500 $$ (initial population in the rural area)
3. $$ U_{n+1} = \frac{1}{2}U_n + \frac{1}{4}R_n $$ (population in the urban area after $$ n+1 $$ years)
4. $$ R_{n+1} = \frac{1}{4}U_n + \frac{3}{4}R_n $$ (population in the rural area after $$ n+1 $$ years)

We can solve this system of linear equations to find the population of each area for any year $$ n \geq 1 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}U_{n+1}=\frac{1}{2}U_{n}+\frac{1}{4}R_{n}\R_{n+1}=\frac{1}{4}U_{n}+\frac{3}{4}R_{n}\U_{1}=300\R_{1}=500\end{array}\right.$$
- step1: Substitute the value of $$U_{1}:$$
  $$\left\{ \begin{array}{l}U_{n+1}=\frac{1}{2}U_{n}+\frac{1}{4}R_{n}\R_{n+1}=\frac{1}{4}U_{n}+\frac{3}{4}R_{n}\R_{1}=500\end{array}\right.$$
- step2: Substitute the value of $$R_{1}:$$
  $$\left\{ \begin{array}{l}U_{n+1}=\frac{1}{2}U_{n}+\frac{1}{4}R_{n}\R_{n+1}=\frac{1}{4}U_{n}+\frac{3}{4}R_{n}\end{array}\right.$$
- step3: Substitute the value of $$U_{n+1}:$$
  $$R_{n+1}=\frac{1}{4}U_{n}+\frac{3}{4}R_{n}$$
- step4: Calculate:
  $$\left(R_{1},R_{n+1},R_{n},U_{1},U_{n+1},U_{n}\right) = \left(500,\frac{1}{4}U_{n}+\frac{3}{4}R_{n},R_{n},300,\frac{1}{2}U_{n}+\frac{1}{4}R_{n},U_{n}\right),\left(U_{n},R_{n}\right) \in \mathbb{R}^{2}$$
- step5: Alternative Form:
  $$\textrm{Infinitely many solutions}$$
The system of linear equations has infinitely many solutions. This means that the population of each area can be expressed in terms of the initial populations and the migration rates, but there are multiple possible solutions depending on the specific migration rates used.

To find the population of each area for any year $$ n \geq 1 $$, we can express the population in the urban area and the rural area in terms of the initial populations and the migration rates. Let's denote the initial population in the urban area as $$ U_0 = 300 $$ and the initial population in the rural area as $$ R_0 = 500 $$.

The population in the urban area after $$ n $$ years can be expressed as:
$$ U_n = \frac{1}{2}U_{n-1} + \frac{1}{4}R_{n-1} $$

The population in the rural area after $$ n $$ years can be expressed as:
$$ R_n = \frac{1}{4}U_{n-1} + \frac{3}{4}R_{n-1} $$

We can use these recursive formulas to calculate the population of each area for any year $$ n \geq 1 $$.
The population in the urban area after $$ n $$ years is $$ U_n = \frac{1}{2}U_{n-1} + \frac{1}{4}R_{n-1} $$, and the population in the rural area is $$ R_n = \frac{1}{4}U_{n-1} + \frac{3}{4}R_{n-1} $$.

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