The debt is amortized by the periodic payment shown. Compute (a) the number of payments required to amortize the debt; (b) the outstanding principal at the time indicated.
Conversion
Period
monthly
Debt Principal Debt Payment
$18,000
$1374
Payment
Interval
1 month
Interest Rate
6%
Outstanding
Principal After:
8th payment
(a) The number of payments required to amortize the debt is
(Round the final answer up to the nearest whole number. Round all intermediate values to six decimal places as needed.)
(b) The outstanding principal is $.
(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

An amortizing loan is the one whose repayment is by a fixed periodic amount, in this case, the repayment( principal and interest) payable per month is $1,374, in order to determine the number of monthly payments required to fully pay off the loan of $18,000, we can make use of the present value formula of an ordinary annuity( since monthly payments would occur at the end of each month) below:

PV=PMT*(1-(1+r)^-N/r

PV=loan amount=$18,000

PMT=monthly payment=$1,374

r=monthly interest rate=6%/12=0.005

N=number of monthly payments=unknown

$18000=$1374*(1-(1+0.005)^-N/0.005

we can rearrange the formula thus:

$18,000/$1374*0.005=1-(1.005)^-N

0.0655021834061135=1-(1.005)^-N

(1.005)^-N=1-0.0655021834061135

(1.005)^-N=0.9344978165938870

take log of both sides

-N*ln(1.005)=ln(0.9344978165938870)

-N=ln(0.9344978165938870)/ln(1.005)

-N=-13.5830425

N=14(to the nearest whole number)

Secondly, the outstanding principal after the 8th payment can be determined using the PV formula above, where is 13.5830425 minus 8 payments already make

PV=$1374*(1-(1+0.005)^-5.5830425 /0.005

PV=$7,546.43

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