1. **Problem statement:** We are given an annuity and need to find: a. How long the annuity will last (in years and months). b. The interest portion of the 27th payment. c. The total interest portion of payments 19 through 30. 2. **Formulas and rules:** For an annuity with regular payments $P$, interest rate per period $i$, and number of periods $n$: - The present value $PV$ is given by: $$PV = P \times \frac{1 - (1+i)^{-n}}{i}$$ - The interest portion of the $k$th payment is: $$\text{Interest}_k = \text{Outstanding balance before payment } k \times i$$ - The principal portion is the payment minus the interest portion. 3. **Step a: How long will the annuity last?** - We need to find $n$ such that the present value matches the loan or investment amount. - Rearranging the formula: $$1 - (1+i)^{-n} = \frac{PV \times i}{P}$$ $$ (1+i)^{-n} = 1 - \frac{PV \times i}{P}$$ $$ -n \ln(1+i) = \ln\left(1 - \frac{PV \times i}{P}\right)$$ $$ n = -\frac{\ln\left(1 - \frac{PV \times i}{P}\right)}{\ln(1+i)}$$ - Calculate $n$ and convert the decimal part to months by multiplying by 12. 4. **Step b: Interest portion of the 27th payment** - Calculate the outstanding balance before the 27th payment: $$\text{Balance}_{26} = P \times \frac{1 - (1+i)^{-(n-26)}}{i}$$ - Then: $$\text{Interest}_{27} = \text{Balance}_{26} \times i$$ 5. **Step c: Interest portion of payments 19 through 30** - Calculate the interest portion for each payment from 19 to 30 and sum them: $$\sum_{k=19}^{30} \text{Interest}_k = \sum_{k=19}^{30} \text{Balance}_{k-1} \times i$$ 6. **Summary:** - Use the formulas above with the given values for $PV$, $P$, and $i$. - Round the annuity length to whole years and months. - Round interest amounts to the nearest cent. Since the problem does not provide specific values for $PV$, $P$, or $i$, the exact numerical answers cannot be computed here. Please provide these values to proceed with calculations.