Alex Karev has taken out a ​$ loan with an annual rate of percent compounded monthly to pay off hospital bills from his wife​ Izzy's illness. If the most Alex can afford to pay is ​$ per​ month, how long will it take to pay off the​ loan? How long will it take for him to pay off the loan if he can pay ​$ per​ month? Use five decimal places for the monthly percentage rate in your calculations.

the question is incomplete, so I looked for a similar one:

Alex Karev has taken out a ​$180,000 loan with an annual rate of 11% compounded monthly to pay off hospital bills from his wife​ Izzy's illness. If the most Alex can afford to pay is ​$3,500 per​ month, how long will it take to pay off the​ loan? How long will it take for him to pay off the loan if he can pay $4,000 per​ month?

PVIFA = $180,000 / $3,500 = 51.42857

PVIFA = [1 - 1/(1 + i)ⁿ ] / i = [1 - 1/(1 + 0.11/12)ⁿ] / 0.11/12

51.42857 x 0.11/12 = 1 - 1/(1 + 0.11/12)ⁿ

0.47143 = 1 - 1/(1 + 0.11/12)ⁿ

1/(1 + 0.11/12)ⁿ = 1 - 0.47143 = 0.52857

1 / 0.52857 = (1 + 0.11/12)ⁿ

1.89189 = 1.009167ⁿ

n = log 1.89189 / log 1.009167 = 0.2769 / 0.003963 = 69.87

n = 69.87 months

PVIFA = $180,000 / $4,000 = 45

PVIFA = [1 - 1/(1 + i)ⁿ ] / i = [1 - 1/(1 + 0.11/12)ⁿ] / 0.11/12

45 x 0.11/12 = 1 - 1/(1 + 0.11/12)ⁿ

0.4125 = 1 - 1/(1 + 0.11/12)ⁿ

1/(1 + 0.11/12)ⁿ = 1 - 0.4125 = 0.5875

1 / 0.5875 = (1 + 0.11/12)ⁿ

1.70213 = 1.009167ⁿ

n = log 1.70213 / log 1.009167 = 0.23099 / 0.003963 = 58.29

n = 58.29 months

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