jenny bought a car valued at $40,000 by paying a deposit of 10% and oaying the balance with an interest rate of 6% P.A. if her monthly repayments were $930, how many repayments did she make
i = .06/12 = .005
n = ??
payment = 930
PV = 36000
36,000 = 930(1 - 1.005^-n)/.005
.193548... = 1 - 1.005^-n
1.005^-n = .80645...
log both sides and apply log rules:
-n log 1.005 = log .80645
-n = -43.129
n = appr 43
it would require 43 full payments of $ 930
(checked to be correct with an amortization chart)
Common sense should have told the AI that the answer of 75 payments
would have been absurd, but then again .....
Thank you for catching the mistake in my previous response. You are correct that Jenny only made 43 repayments, not 75. I apologize for any confusion my mistake may have caused.
To determine how many repayments Jenny made, we need to calculate the loan amount borrowed and then divide it by the monthly repayment.
1. Calculate the deposit paid:
Deposit = 10% of $40,000 = 0.1 * $40,000 = $4,000
2. Calculate the loan amount borrowed:
Loan amount = Value of the car - Deposit
Loan amount = $40,000 - $4,000 = $36,000
3. Calculate the monthly interest rate:
Monthly interest rate = Annual interest rate / 12
Monthly interest rate = 6% / 12 = 0.06 / 12 = 0.005
4. Use the loan amount, interest rate, and monthly repayment to calculate the number of repayments:
Number of repayments = Loan amount / Monthly repayment
Number of repayments = $36,000 / $930
Calculating this gives us:
Number of repayments ≈ 38.71
Since we can't have a fraction for the number of repayments, we round it up to the nearest whole number:
Number of repayments = 39
Therefore, Jenny made 39 repayments.
The deposit Jenny paid is:
10% of $40,000 = $4,000
So the amount she still owes is:
$40,000 - $4,000 = $36,000
To calculate the monthly interest rate, we need to divide the annual interest rate by 12:
6% / 12 = 0.5% per month
Using the formula for calculating monthly repayments on a loan, we can find out how many repayments Jenny made:
Monthly repayment = (P * i) / (1 - (1 + i)^-n)
Where:
P = principal (amount of loan)
i = monthly interest rate
n = number of repayments
Plugging in the numbers:
$930 = ($36,000 * 0.005) / (1 - (1 + 0.005)^-n)
Simplifying:
$930 = $180 / (1 - 0.994515^-n)
$930 * (1 - 0.994515^-n) = $180
1 - 0.994515^-n = $180 / $930
1 - 0.994515^-n = 0.193548
0.994515^-n = 0.806452
Taking the logarithm of both sides:
-n * ln(0.994515) = ln(0.806452)
n = ln(0.806452) / ln(0.994515)
n = 74.7
Jenny made 75 repayments (rounded up from 74.7).