Certainly! Let's break down the solution to the problem into two parts.
### Part (a): Calculate the overall percent inflation during the period
The formula to calculate the percent inflation over a period is:
[tex]\[ \text{Percent Inflation} = \left(\frac{\text{CPI in Final Year} - \text{CPI in Initial Year}}{\text{CPI in Initial Year}}\right) \times 100 \][/tex]
Given:
- CPI in 2010 = 116.5
- CPI in 2015 = 126.6
Substitute these values into the formula:
[tex]\[ \text{Percent Inflation} = \left(\frac{126.6 - 116.5}{116.5}\right) \times 100 \][/tex]
First, find the difference in CPI:
[tex]\[ 126.6 - 116.5 = 10.1 \][/tex]
Next, divide this difference by the CPI in 2010:
[tex]\[ \frac{10.1}{116.5} \approx 0.0867 \][/tex]
Finally, multiply by 100 to express it as a percentage:
[tex]\[ 0.0867 \times 100 \approx 8.67\% \][/tex]
So, the overall percent inflation during the period from 2010 to 2015 is approximately 8.67%.
### Part (b): Calculate the required salary in 2015 to keep pace with inflation
To find out how much a worker earning [tex]$50,000 in 2010 would need to earn in 2015 to keep pace with inflation, we use the ratio of the CPI values from the two years.
Given:
- Salary in 2010 = $[/tex]50,000
- CPI in 2010 = 116.5
- CPI in 2015 = 126.6
The formula to adjust the salary is:
[tex]\[ \text{Adjusted Salary} = \text{Salary in Initial Year} \times \left(\frac{\text{CPI in Final Year}}{\text{CPI in Initial Year}}\right) \][/tex]
Substitute the values:
[tex]\[ \text{Adjusted Salary} = 50{,}000 \times \left(\frac{126.6}{116.5}\right) \][/tex]
First, compute the ratio of the CPIs:
[tex]\[ \frac{126.6}{116.5} \approx 1.0867 \][/tex]
Next, multiply this ratio by the salary in 2010:
[tex]\[ 50{,}000 \times 1.0867 \approx 54{,}335 \][/tex]
So, to keep pace with inflation, a worker who earned [tex]$50,000 in 2010 would need to earn approximately $[/tex]54,335 in 2015.
