1. **Problem Statement:** Cindy borrows money and repays with monthly payments of 889 at the beginning of every month. Interest rate is 9.902% per year, compounded monthly, and loan is paid off after 4 payments. We want to find how much she owes immediately before payment 3. 2. **Given Data:** - Monthly payment, $P = 889$ - Annual nominal interest rate, $i_{annual} = 9.902\% = 0.09902$ - Number of payments, $n = 4$ - Payments are made at the beginning of the month (annuity due) 3. **Calculate monthly interest rate:** $$ i = \frac{0.09902}{12} = 0.0082517 $$ 4. **Relevant formula:** Loan amount $L$ satisfies the present value of an annuity due: $$ L = P \times \frac{1 - (1+i)^{-n}}{i} \times (1+i) $$ 5. **Calculate loan amount:** $$ L = 889 \times \frac{1-(1.0082517)^{-4}}{0.0082517} \times 1.0082517 $$ Calculate each part: $$ (1.0082517)^{-4} = \frac{1}{(1.0082517)^4} = \frac{1}{1.0334} = 0.9677 $$ $$ 1 - 0.9677 = 0.0323 $$ $$ \frac{0.0323}{0.0082517} = 3.9151 $$ $$ L = 889 \times 3.9151 \times 1.0082517 = 889 \times 3.9476 = 3508.02 $$ Loan amount is approximately $3508.02$. 6. **Find amount owed immediately before payment 3:** After payments 1 and 2, the principal will reduce, but we want the balance just before payment 3, so after interest accrual on the previous balance, but before payment 3 is made. 7. **Step 1: Calculate balance immediately after payment 2 (start of month 3):** Using annuity due, the balance after $k$ payments at beginning of periods is: $$ B_k = L \times (1+i)^k - P \times \frac{(1+i)^k - 1}{i} $$ For $k=2$: $$ B_2 = 3508.02 \times (1.0082517)^2 - 889 \times \frac{(1.0082517)^2 -1}{0.0082517} $$ Calculate powers: $$ (1.0082517)^2 = 1.01658 $$ Calculate numerator for payment term: $$ 1.01658 -1 = 0.01658 $$ Payment factor: $$ \frac{0.01658}{0.0082517} = 2.0094 $$ Calculate: $$ 3508.02 \times 1.01658 = 3566.17 $$ $$ 889 \times 2.0094 = 1786.36 $$ Balance before payment 3: $$ B_2 = 3566.17 - 1786.36 = 1779.81 $$ 8. **Interpretation:** Immediately before payment 3, Cindy owes about 1779.81. **Final Answer: $1779.81$