Aliyah bought a house with a mortgage of $419,800. The mortgage is being financed with an interest rate of 4.91% compounded monthly. Aliyah will make payments of $1,836.
(a) How many payments will Aliyah have to make to repay the mortgage?
payment(s)
(b) How long, in months, will it take Aliyah to pay off the mortgage? (Hint: In an annuity due, payments are made at the beginning of each period.)
months)

To solve this problem, we need to use the formula for the number of payments on a mortgage with monthly compounding interest.

The formula for the monthly mortgage payment $ M $ is derived from the present value of an annuity:

\[
M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]

Where:
- $ M $ is the monthly payment,
- $ P $ is the loan amount (mortgage principal),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of payments.

Given:
- $ P = 419,800 $,
- The annual interest rate is 4.91%, so the monthly interest rate $ r = \frac{4.91\%}{12} = 0.00409167 $,
- The monthly payment $ M = 1,836 $.

We need to solve for $ n $, the number of payments Aliyah will make. To do this, we rearrange the formula to solve for $ n $:

\[
n = \frac{\log\left(\frac{M}{M - P \times r}\right)}{\log(1 + r)}
\]

I'll calculate $ n $, then determine how many months it will take Aliyah to pay off the mortgage.(a) Aliyah will need to make approximately **672 payments** to fully repay the mortgage.

(b) Since the payments are monthly, it will take her **672 months** to pay off the mortgage.

Would you like further details or clarifications?

Here are 5 related questions to explore:
1. How does the interest rate affect the number of payments?
2. What happens if Aliyah increases her monthly payments by $100?
3. How long would it take to pay off the loan with a different interest rate, say 5.5%?
4. How much total interest will Aliyah pay over the life of the loan?
5. What is the difference between an annuity due and an ordinary annuity?

**Tip:** Increasing the frequency or amount of payments can significantly reduce the total interest paid over the mortgage term.

Aliyah bought a house with a mortgage of $419,800. The mortgage is being financed with an interest rate of 4.91% compounded monthly. Aliyah will make payments of $1,836.
(a) How many payments will Aliyah have to make to repay the mortgage?
(b) How long, in months, will it take Aliyah to pay off the mortgage?

Discover how to calculate the number of payments required to repay a mortgage of $419,800 at a 4.91% interest rate, with monthly payments of $1,836. This detailed solution covers key concepts in annuities, compound interest, and logarithms. Suitable for advanced high school or college-level students.

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