# GMAT Challenge Series (700+): Q19

Question:

Louie takes out a three-month loan of $1000. The lender charges him 10% interest per month compounded monthly. The terms of the loan state that Louie must repay the loan in three equal monthly payments. To the nearest dollar, how much does Louie have to pay each month? A.$333

B. $383 C.$402

D. $433 E.$483

$$\text{Let the monthly repayment be: } \textit{x}$$  $$\text{Month 0:}$$ $$\textit{Unpaid Balance} = {1,000}$$  $$\text{Month 1:}$$ $$\textit{Unpaid Balance} = \left({{1} + \textit{Interest Rate}}\right) \left(\textit{Unpaid Balance Month 0}\right) - \textit{Monthly Repayment}$$ $$\textit{Unpaid Balance} = \left({1.1}\right) \left({1,000}\right) - \textit{x}$$ $$\textit{Unpaid Balance} = {1,100} - \textit{x}$$  $$\text{Month 2:}$$ $$\textit{Unpaid Balance} = \left({{1} + \textit{Interest Rate}}\right) \left(\textit{Unpaid Balance Month 1}\right) - \textit{Monthly Repayment}$$ $$\textit{Unpaid Balance} = \left({1.1}\right) \left({{1,100} - \textit{x}}\right) - \textit{x}$$ $$\textit{Unpaid Balance} = {1,210} - {2.1}\textit{x}$$  $$\text{Month 3:}$$ $$\textit{Unpaid Balance} = \left({{1} + \textit{Interest Rate}}\right) \left(\textit{Unpaid Balance Month 2}\right) - \textit{Monthly Repayment}$$ $$\textit{Unpaid Balance} = \left({1.1}\right) \left({{1,210} - {2.1}\textit{x}}\right) - \textit{x}$$ $$\textit{Unpaid Balance} = {1,331} - {3.31}\textit{x}$$  $$\text{At the end of Month 3 the loan is repaid, therefore:}$$ $${0} = {1,331} - {3.31}\textit{x}$$ $$\textit{x} \approx {402}$$