Each year 4% of people living in Tauranga move to Christchurch, and 4% of people in Christchurch move to Tauranga. If the populations of Christchurch and Tauranga are currently 381,877 and 131,887 people, respectively, what was the population of Christchurch last year (to the nearest individual)?

You must use matrices to solve this question, as other methods may lead to premature rounding.

Question

Each year 4% of people living in Tauranga move to Christchurch, and 4% of people in Christchurch move to Tauranga. If the populations of Christchurch and Tauranga are currently 381,877 and 131,887 people, respectively, what was the population of Christchurch last year (to the nearest individual)?

You must use matrices to solve this question, as other methods may lead to premature rounding.

BlackTom AI · Accepted Answer

Model the populations by the state vector x_t = [C_t; T_t] and the yearly transfer as a linear map x_{t+1} = A x_t with
A = [[0.96, 0.04],
     [0.04, 0.96]].
(Each year Christchurch keeps 96% of its people and receives 4% of Tauranga, and vice versa.)

1) Relevant knowledge points
- Discrete linear dynamical systems: x_{t+1} = A x_t.
- To find last year’s population x_{t-1} given x_t, compute x_{t-1} = A^{-1} x_t.
- Inverse of a 2×2 matrix [[a,b],[b,a]] is 1/(a^2-b^2) * [[a,-b],[-b,a]]; working with exact fractions avoids premature rounding.

2) Step-by-step reasoning and 3) intermediate calculations
Given current populations x_t = [381,877; 131,887], set a = 0.96 = 24/25 and b = 0.04 = 1/25.
det(A) = a^2 − b^2 = (24/25)^2 − (1/25)^2 = (576−1)/625 = 575/625 = 23/25.

A^{-1} = 1/det(A) * [[a, −b], [−b, a]]
       = (25/23) * [[24/25, −1/25], [−1/25, 24/25]]
       = [[24/23, −1/23], [−1/23, 24/23]].

Last year’s Christchurch population (first component of x_{t-1}) is
(24/23)*381,877 − (1/23)*131,887
= (24·381,877 − 131,887)/23
= (9,165,048 − 131,887)/23
= 9,033,161 / 23
= 392,746 + 3/23
≈ 392,746.13043478.

Rounding to the nearest individual gives 392,746.

4) Why this is correct
Using the inverse of the exact transition matrix yields the exact rational expression 9,033,161/23, avoiding intermediate decimal rounding. That produces the precise prior-year population; any approximate inversion or premature rounding could shift the final integer by one.

5) Other options (if any)
No multiple-choice options were provided. A direct algebraic approach (solving two linear equations) is equivalent but must keep exact arithmetic; numeric rounding before inversion can produce small errors.

6) Conclusion
The population of Christchurch last year was 392,746, matching the provided answer.

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