IB Mathematics Applications & Interpretation SL 1.7—loan Repayments and Amortization Questionbank

Find the amount of each monthly payment.

Show that the amount owing immediately after the $24$th payment is $\text{＄}60219.51$.

Calculate the interest part of the $25$th payment, the amount by which that payment reduces the amount owing, and the percentage of that payment which is interest. The members meet immediately before the $85$th payment.

Find the interest part of the $85$th payment and the amount by which it reduces the amount owing. Compare this split with the split found in part (c). The treasurer must report to the meeting on the five years covered by the $25$th payment up to and including the $84$th.

Find the total interest charged on the loan over those sixty payments. The club's constitution requires the amount owing to fall below half the amount borrowed before the end of the sixth year of the loan. If it did not, the members must vote at this meeting on an increase in subscriptions.

Find the number of the first payment after which the amount owing is less than half the amount borrowed. Deduce whether the constitution's requirement was met.

Assuming the same monthly payments but no interest, calculate the amount still owing immediately after payment $84$. Explain what the $\text{＄}10501.78$ calculated in part (e) represents and why it does not reduce the principal.

Calculate the interest rate per quarter, and show that a quarterly payment of $\text{＄}1024$ leaves the amount owing unchanged at $\text{＄}64000$. The members are not certain what the first four harvests will bring. One of them asks what would happen if the co-operative could pay only $\text{＄}900$ at the end of each of the first four quarters.

Find the amount that would be owing at the start of the second year, and explain what a payment below the interest charged does to the amount owing. The bank and the co-operative agree that the reduced payment for each of the first four quarters will be $\text{＄}1024$, so that the amount owing at the start of the second year is $\text{＄}64000$.

Find the quarterly payment that clears this amount in the $24$ payments of the second year onwards.

Calculate the total amount the co-operative pays over the whole seven-year arrangement, and the total interest charged. The pump is installed as planned, but the third harvest is poor. The co-operative pays only $\text{＄}700$ at the end of the third quarter, and the payments at the end of the first, second and fourth quarters are each $\text{＄}1024$ as agreed. The $24$ later payments must still clear whatever is owing at the start of the second year.

Find the amount owing at the start of the second year, and the revised quarterly payment needed to clear it in $24$ payments. The bank says it would instead have accepted $28$ equal quarterly payments, the first of them at the end of the very first quarter, clearing the loan over the same seven years with no reduced-payment year at all.

Find this quarterly payment and the total interest charged under that plan, and compare this total with the total interest found in part (d). Every payment worked out so far has assumed that the nominal rate of $6.4\ \%$ holds for all seven years. In fact the agreement fixes that rate for the first year only: at the start of the second year the bank sets the nominal rate again, and whatever rate it sets then applies until the loan is cleared.

Explain how an increase or decrease in the reviewed interest rate would affect the remaining quarterly payments. Explain why the review does not change the balance immediately before the new rate takes effect.