Of two-car families in a small city, 88% remain two-car families in the following year and 12% become one-car families in the following year. Of one-car families, 72% remain one-car families and 28% become two-car families. Suppose these trends continue for a few years. At present, 4800 families have one car and 4200 families have two cars.
Find the numbers of one-car familes and two car families two years from now.
I assume you are learning about Markov's chains in the study of matrices
end of year 1:
[4200 4800] x
.88 .12
.72 .28
=
[7152 1848]
end of year 2
[7152 1848]
.88 .12
.72 .28
=
[7624.32 1375.68]
I am not absolutely sure if I have the correct order, haven't done this kind of math in almost 20 years.
To find the number of one-car and two-car families two years from now, we can use the given percentages and the current number of families with one and two cars.
Let's start by finding the number of one-car families two years from now:
1. First, we need to determine how many of the current one-car families will remain one-car families in the following year. Given that 72% of one-car families remain one-car families, we can calculate it as follows:
Number of one-car families remaining = 72% of current one-car families
Number of one-car families remaining = 0.72 * 4800
2. Next, we need to determine how many of the current two-car families will become one-car families in the following year. Given that 12% of two-car families become one-car families, we can calculate it as follows:
Number of two-car families becoming one-car families = 12% of current two-car families
Number of two-car families becoming one-car families = 0.12 * 4200
3. Now, let's calculate the total number of one-car families two years from now by summing up the remaining one-car families and the two-car families becoming one-car families:
Total number of one-car families two years from now = Number of one-car families remaining + Number of two-car families becoming one-car families
Now, let's find the number of two-car families two years from now:
4. We need to determine how many of the current one-car families will become two-car families in the following year. Given that 28% of one-car families become two-car families, we can calculate it as follows:
Number of one-car families becoming two-car families = 28% of current one-car families
Number of one-car families becoming two-car families = 0.28 * 4800
5. We also need to determine how many of the current two-car families will remain two-car families in the following year. Given that 88% of two-car families remain two-car families, we can calculate it as follows:
Number of two-car families remaining = 88% of current two-car families
Number of two-car families remaining = 0.88 * 4200
6. Finally, let's calculate the total number of two-car families two years from now by summing up the one-car families becoming two-car families and the remaining two-car families:
Total number of two-car families two years from now = Number of one-car families becoming two-car families + Number of two-car families remaining
Now we can calculate the numbers of one-car and two-car families two years from now:
Total number of one-car families two years from now = (0.72 * 4800) + (0.12 * 4200)
Total number of two-car families two years from now = (0.28 * 4800) + (0.88 * 4200)