To determine the overall increase in prices due to inflation from 2000 to 2013 and the annual inflation rate assuming annual compounding, we can follow these steps:
### Part (a): Overall Increase in Prices Due to Inflation
1. Identify the CPI values for the starting and ending years:
- Start Year (2000): [tex]\( \text{CPI}_{2000} = 172.2 \)[/tex]
- End Year (2013): [tex]\( \text{CPI}_{2013} = 233.0 \)[/tex]
2. Calculate the overall increase in CPI from 2000 to 2013:
[tex]\[
\text{Overall Increase} = \frac{\text{CPI}_{2013} - \text{CPI}_{2000}}{\text{CPI}_{2000}} \times 100
\][/tex]
3. Substitute the values into the formula and compute:
[tex]\[
\text{Overall Increase} = \frac{233.0 - 172.2}{172.2} \times 100
\][/tex]
[tex]\[
\text{Overall Increase} = \frac{60.8}{172.2} \times 100
\][/tex]
4. Simplify and round to one decimal place:
[tex]\[
\text{Overall Increase} \approx 35.3\%
\][/tex]
Therefore, the overall increase in prices due to inflation from 2000 to 2013 is 35.3%.
### Part (b): Annual Inflation Rate Assuming Annual Compounding
1. Calculate the number of years between 2000 and 2013:
[tex]\[
\text{Number of Years} = 2013 - 2000 = 13
\][/tex]
2. Use the formula for the annual inflation rate with annual compounding, which is derived from the compounded interest formula:
[tex]\[
\left( \frac{\text{CPI}_{2013}}{\text{CPI}_{2000}} \right)^{\frac{1}{\text{Number of Years}}} - 1
\][/tex]
3. Substitute the values into the formula and compute:
[tex]\[
\text{Annual Inflation Rate} = \left( \frac{233.0}{172.2} \right)^{\frac{1}{13}} - 1
\][/tex]
4. Convert the result to a percentage and round to one decimal place:
[tex]\[
\text{Annual Inflation Rate} \approx 2.4\%
\][/tex]
Therefore, the annual inflation rate, assuming annual compounding, from 2000 to 2013 is 2.4%.
