Certainly. Let me guide you step-by-step through this problem:
1. Understanding the Growth Rates:
- Town X's population increases by 8% per year.
- Town Y's population increases by 12% per year.
2. Establish the Time Frame:
- We are examining the change from the beginning of 2010 to the end of 2012, which is 3 years.
3. Express the Growth Mathematically:
- The formula for population growth over a number of years with a constant rate is:
[tex]\[
P_{\text{final}} = P_{\text{initial}} \times (1 + \text{rate})^{\text{number of years}}
\][/tex]
4. Equating Final Populations:
- Let [tex]\( P_X \)[/tex] be the population of Town X at the beginning of 2010.
- Let [tex]\( P_Y \)[/tex] be the population of Town Y at the beginning of 2010.
- At the end of 2012, the populations are equal:
[tex]\[
P_X \times (1 + 0.08)^3 = P_Y \times (1 + 0.12)^3
\][/tex]
5. Calculate the Growth Factors:
- For Town X over 3 years:
[tex]\[
\text{Growth Factor for X} = (1 + 0.08)^3 \approx 1.259712
\][/tex]
- For Town Y over 3 years:
[tex]\[
\text{Growth Factor for Y} = (1 + 0.12)^3 \approx 1.404928
\][/tex]
6. Establish the Relationship Between Initial Populations:
- Since the final populations are equal, the scaled relationship is:
[tex]\[
P_X \times 1.259712 = P_Y \times 1.404928
\][/tex]
- Rearrange to find the ratio:
[tex]\[
\frac{P_X}{P_Y} = \frac{1.404928}{1.259712} \approx 1.115277
\][/tex]
Therefore, the ratio of the population of Town X to the population of Town Y at the beginning of 2010 is approximately [tex]\( 1.115277 \)[/tex]. This means that the initial population of Town X was about 11.53% greater than that of Town Y at the beginning of 2010.
