Let's solve the problem step by step: --- ## Given: - **Future Value (FV):** \$84,600 (amount due in 10 years) - **Number of years (n):** 10 - **Annual interest rate (r):** 10% (compound interest) - **Payments:** - Part (a): **Annual** at the end of each year - Part (b): **Monthly** at the end of each month --- ## a. **What will the annual payments have to be?** We use the **future value of an ordinary annuity** formula: \[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) = Future Value (\$84,600) - \( PMT \) = Annual payment (what we want to find) - \( r \) = Interest rate per period (10% = .10) - \( n \) = Number of periods (10 years) **Rearrange to solve for \( PMT \):** \[ PMT = \frac{FV \times r}{(1 + r)^n - 1} \] **Plug in the values:** \[ PMT = \frac{84,600 \times .10}{(1.10)^{10} - 1} \] Calculate \( (1.10)^{10} \): \[ (1.10)^{10} = 2.5937424601 \] So, \[ PMT = \frac{84,600 \times .10}{2.5937424601 - 1} \] \[ PMT = \frac{8,460}{1.5937424601} \] \[ PMT = 5,310.01 \] **Final Answer (rounded to nearest whole number):** \[ \boxed{5,310} \] --- ## b. **If payments are made monthly, how much would they need to pay each month?** Now, payments are monthly, so: - \( n = 10 \times 12 = 120 \) months - \( r = 10\%/12 = .10/12 = .008333 \) per month Formula (future value of an ordinary annuity): \[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \] \[ PMT = \frac{FV \times r}{(1 + r)^n - 1} \] Plug in the values: \[ (1 + r)^n = (1.008333)^{120} = 2.70704 \] \[ PMT = \frac{84,600 \times .008333}{2.70704 - 1} \] \[ PMT = \frac{705.00}{1.70704} \] \[ PMT = 412.97 \] **Final Answer (rounded to nearest whole number):** \[ \boxed{413} \] --- ## c. **How much less will the borrower pay each year with monthly payments versus annual payments?** Calculate **total payment per year** for both cases: - **Annual payments:** \( 5,310 \) per year - **Monthly payments:** \( 413 \times 12 = 4,956 \) per year **Difference:** \[ 5,310 - 4,956 = 354 \] **Final Answer:** \[ \boxed{354} \] --- ## **Summary Table** | Payment Type | Payment Amount | Total per Year | |--------------|----------------|---------------| | Annual | \$5,310 | \$5,310 | | Monthly | \$413 | \$4,956 | **Borrower pays \$354 less per year with monthly payments.** --- If you need a more detailed explanation or breakdown of calculations, let me know!