Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 04 (b)

MATH152 April 2016

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Question B 04 (b)
Suppose in the year 2020, 50 million people live in cities and 50 million in the suburbs. Every year, 10% of city residents move to the suburbs and 20% of the residents of the suburbs move to cities. (b) What fraction of residents will be living in cities in 2022?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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Hint
We use matrix $P$ in part (a), which shows the transition probability for each year.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The matrix $P={\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}$ shows the transition of each year. The initial population of city and suburb can be written as ${\begin{pmatrix}50&50\end{pmatrix}}^{T}.$ Since we need to find the population of city in 2022 ( two years after 2020), we compute the following: ${\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}{\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}{\begin{pmatrix}50\\50\end{pmatrix}}={\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}{\begin{pmatrix}55\\45\end{pmatrix}}={\begin{pmatrix}58.5\\41.5\end{pmatrix}}.$ So, the population of city is 58.5 million in 2022. So, the fraction is ${\frac {58.5}{100}}=0.585.$