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?

Answer 1

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)