Certainly! Let's solve this problem step-by-step:
### Problem Analysis
1. Town X: Decreases by 8% per year.
2. Town Y: Increases by 12% per year.
3. At the end of 2012, the populations of the two towns are equal.
4. We need to determine the ratio of the population of town X to the population of town Y at the beginning of 2010.
### Step-by-Step Solution
1. Initial Populations:
- Let the population of town X at the beginning of 2010 be [tex]\( P_X \)[/tex].
- Let the population of town Y at the beginning of 2010 be [tex]\( P_Y \)[/tex].
2. Population Changes Over Three Years:
- Town X decreases by 8% each year, which means it retains 92% [tex]\((100\% - 8\% = 92\%)\)[/tex] of its population each year.
- Town Y increases by 12% each year, which means it becomes 112% [tex]\((100\% + 12\% = 112\%)\)[/tex] of its population each year.
3. Population Formulas After Three Years:
- For town X:
[tex]\[
P_{X\_2012} = P_X \times 0.92^3
\][/tex]
- For town Y:
[tex]\[
P_{Y\_2012} = P_Y \times 1.12^3
\][/tex]
4. Equality at the End of 2012:
- At the end of 2012, the populations are equal:
[tex]\[
P_X \times 0.92^3 = P_Y \times 1.12^3
\][/tex]
5. Ratio of Initial Populations:
- To find the ratio [tex]\( \frac{P_X}{P_Y} \)[/tex], we rearrange the equation:
[tex]\[
\frac{P_X}{P_Y} = \frac{1.12^3}{0.92^3}
\][/tex]
6. Calculating the Ratio:
- The calculation yields:
[tex]\[
\frac{1.12^3}{0.92^3} \approx 1.804
\][/tex]
### Conclusion
The ratio of the population of town X to the population of town Y at the beginning of 2010 is approximately [tex]\( 1.804 \)[/tex].
Hence, if the population of town Y at the beginning of 2010 was one unit, the population of town X at the same time would be approximately 1.804 units.
