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a debt of ​$ is repaid by making payments of​ $3500. if interest is​ 9% compounded​ monthly, for how long will payments have to be​ made at the end of every three months with payments deferred for five​ years?

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The time it will take for payments to be made at the end of every three months, with payments deferred for five years is: 7 years and 9 months

How to find the repayment amount period?

If payments are made at the end of every three months with payments deferred for five years:

In this case, we need to calculate the future value of the annuity with quarterly payments and compound interest, while considering the five-year deferral period.

Future Value = $20,000

Payment = $3,500

Interest Rate per Period (r) = 9% / 12 = 0.75% (monthly interest rate)

Number of Periods (n) = ?

Since there is a deferral period of five years, we need to subtract those five years from the total number of periods:

Total Number of Periods = Number of Periods (deferred) + Number of Periods (repayment)

To calculate the number of periods (repayment), we use the formula:

Number of Periods (repayment) = (log(1 + (Future Value × r) / Payment)) / log(1 + r)

Substituting the given values, we have:

Number of Periods (repayment) = (log(1 + ($20,000 × 0.0075) / $3,500)) / log(1 + 0.0075)

Calculating this, we find:

Number of Periods (repayment) ≈ 11.56 periods

Subtracting the deferral period, we have:

Number of Periods (deferred) = 5 years × 4 quarters per year = 20 quarters

Total Number of Periods = Number of Periods (deferred) + Number of Periods (repayment)

Total Number of Periods = 20 + 11.56 ≈ 31.56 periods

Therefore, payments will have to be made for approximately 31.56 quarters (or about 7 years and 9 months) at the end of every three months, with payments deferred for five years.

Complete question is:

A debt of $20000 is repaid by making payments of $3500. if interest is 9% compounded monthly, for how long will payments have to be made at the end of every three months with payments deferred for five years?

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