Payments of $1800 and $2400 weere made on a $10,000 variable-rate loan 18 and 30 months after the date of the loan. The interest rate was 11.5% compounded semi-annually for the first two years and 10.74% compounded monthly thereafter. What amount was owed on the loan after three years?

make a 'time-graph' and label points 0,(now), 18,24 (the rate changes here), 30, and 36.

place 10000 at 0
1800 at 18 and 2400 at 30

you want to bring all three money values up to time-point 36

the 1800 is charged a rate of .115/12 for 6 months and then .1074/12 for the next 12 months.
its value at 36 = 1800(1+.115/12)^6(1+.1074/12)^12
= $2121.10

the 2400 is charged .1074/12 for 6 months
its value is 2400(1+.1074/12)^6
= $2531.80

The original debt at 36 has a value of
10000(1+.115/12)^24(1+.1074/12)^12
= 13990.97

so the amount owing is
13990.97 - 2531.80 - 2121.10
= 9338.07

(please check my arithmetic)

To calculate the amount owed on the loan after three years, we can break down the problem into two parts: the first two years, during which the interest rate is 11.5% compounded semi-annually, and the remaining year, during which the interest rate is 10.74% compounded monthly.

First, let's calculate the amount owed after the first two years using the semi-annually compounded interest formula:

A = P(1 + r/n)^(nt)

Where:
A = the future value or amount owed after the specified time
P = the principal amount (initial loan balance)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = time in years

For the first two years:
P = $10,000
r = 11.5% or 0.115 (in decimal form)
n = 2 (compounded semi-annually)
t = 2 years

A = 10000(1 + 0.115/2)^(2*2)
A ≈ $11,973.62

After two years, the amount owed is approximately $11,973.62.

Next, we need to calculate the amount owed after the remaining one year using the monthly compounded interest formula:

A = P(1 + r/n)^(nt)

For the remaining one year:
P = $11,973.62 (the amount owed after two years)
r = 10.74% or 0.1074 (in decimal form)
n = 12 (compounded monthly)
t = 1 year

A = 11973.62(1 + 0.1074/12)^(12*1)
A ≈ $12,685.56

After three years, the amount owed on the loan is approximately $12,685.56.