Sure! Let's work through the problem to determine the ratio of the population of town X (Px) to the population of town Y (Py) at the beginning of 2010.
### Step-by-Step Solution:
1. Understand the problem setup:
- Town X's population decreases by 8% per year from 2010 to 2012.
- Town Y's population increases by 12% per year from 2010 to 2012.
- Populations of both towns are equal at the end of 2012.
2. Define the population changes:
- For town X: If the population at the beginning of 2010 is Px, then at the end of each year, it decreases to 92% of the previous year. Thus, at the end of 2012, it is:
[tex]\[
\text{Population of X at end of 2012} = Px \times (0.92^3)
\][/tex]
- For town Y: If the population at the beginning of 2010 is Py, then at the end of each year, it increases to 112% of the previous year. Thus, at the end of 2012, it is:
[tex]\[
\text{Population of Y at end of 2012} = Py \times (1.12^3)
\][/tex]
3. Equate the populations at the end of 2012:
- Since the populations are equal at the end of 2012:
[tex]\[
Px \times (0.92^3) = Py \times (1.12^3)
\][/tex]
4. Solve for [tex]\( \frac{Px}{Py} \)[/tex]:
- Rearrange the equation to solve for the ratio of [tex]\( Px \)[/tex] to [tex]\( Py \)[/tex] at the beginning of 2010:
[tex]\[
\frac{Px}{Py} = \frac{1.12^3}{0.92^3}
\][/tex]
5. Calculate the ratio:
- By calculating the above ratio, we get:
[tex]\[
\frac{Px}{Py} = 1.804224541793376
\][/tex]
### Conclusion:
The ratio of the population of town X (Px) to the population of town Y (Py) at the beginning of 2010 is approximately [tex]\( \boxed{1.804224541793376} \)[/tex].
