Chapter 9.3, Problem 61E

Loans. A credit union classifies car loans into one of four categories: the loan has been paid in full $\left(F\right)$. the account is in good standing $\left(G\right)$ with all payments up to date, the account is in arrears $\left(A\right)$ with one or more missing payments, or the account has been classified as a bad debt $\left(B\right)$ and sold to a collection agency. Past records indicate that each month $10\%$ of the accounts in good standing pay the loan in full, $80\%$ remain in good standing, and $10\%$ become in arrears. Furthermore, $10\%$ of the accounts in arrears are paid in full, $40\%$ become accounts in good standing, $40\%$ remain in arrears, and $10\%$ are classified as bad debts.

(A) In the long run, what percentage of the accounts in arrears will pay their loan in full?

(B) In the long run, what percentage of the accounts in good standing will become bad debts?

(C) What is the average number of months that an account in arrears will remain in this system before it is either paid in full or classified as a bad debt?