Answer :
Certainly! Let's determine the salary you would need in 2017 to have the same purchasing power as a $70,000 salary in 2013, given an annual inflation rate of 9%. We will use the compound interest formula, which is typically used to calculate future values under constant growth rates, for our computation.
Here are the steps:
1. Identify key values:
- Initial salary in 2013 (\( P \)): $70,000
- Annual inflation rate (\( r \)): 9% or 0.09
- Number of years (\( t \)) from 2013 to 2017: 2017 - 2013 = 4 years
2. Understand the compound interest formula:
The formula for future value considering compound interest is:
[tex]\[ A = P \times (1 + r)^t \][/tex]
where:
- \( A \) is the future value or the salary needed in 2017.
- \( P \) is the initial principal balance (salary in 2013).
- \( r \) is the annual interest rate (inflation rate).
- \( t \) is the number of years.
3. Substitute the values into the formula:
[tex]\[ A = 70000 \times (1 + 0.09)^4 \][/tex]
4. Calculate the amount needed:
First, compute \( (1 + 0.09) \):
[tex]\[ 1 + 0.09 = 1.09 \][/tex]
Then, raise this amount to the power of 4:
[tex]\[ 1.09^4 \approx 1.411582 \][/tex]
Now, multiply this result by the initial salary:
[tex]\[ 70000 \times 1.411582 \approx 98810.7127 \][/tex]
5. Round to two decimal places:
[tex]\[ 98810.7127 \approx 98810.71 \][/tex]
So, to maintain the same purchasing power in 2017 with a 9% annual inflation rate, you would need a salary of approximately $98,810.71.
Thus, the required salary in 2017 to have the same purchasing power as [tex]$70,000 in 2013, given an annual inflation rate of 9%, is $[/tex]98,810.71.
Here are the steps:
1. Identify key values:
- Initial salary in 2013 (\( P \)): $70,000
- Annual inflation rate (\( r \)): 9% or 0.09
- Number of years (\( t \)) from 2013 to 2017: 2017 - 2013 = 4 years
2. Understand the compound interest formula:
The formula for future value considering compound interest is:
[tex]\[ A = P \times (1 + r)^t \][/tex]
where:
- \( A \) is the future value or the salary needed in 2017.
- \( P \) is the initial principal balance (salary in 2013).
- \( r \) is the annual interest rate (inflation rate).
- \( t \) is the number of years.
3. Substitute the values into the formula:
[tex]\[ A = 70000 \times (1 + 0.09)^4 \][/tex]
4. Calculate the amount needed:
First, compute \( (1 + 0.09) \):
[tex]\[ 1 + 0.09 = 1.09 \][/tex]
Then, raise this amount to the power of 4:
[tex]\[ 1.09^4 \approx 1.411582 \][/tex]
Now, multiply this result by the initial salary:
[tex]\[ 70000 \times 1.411582 \approx 98810.7127 \][/tex]
5. Round to two decimal places:
[tex]\[ 98810.7127 \approx 98810.71 \][/tex]
So, to maintain the same purchasing power in 2017 with a 9% annual inflation rate, you would need a salary of approximately $98,810.71.
Thus, the required salary in 2017 to have the same purchasing power as [tex]$70,000 in 2013, given an annual inflation rate of 9%, is $[/tex]98,810.71.
