Question: Decreasing the number of years of a loan decreases the amount of interest repaid over the term of the loan. Suppose a dental hygienist has the option of a 30-year loan or a 25-year loan of $355,000 at an annual interest rate of 4.25%. (a) Calculate the monthly payment (in dollars) for each loan. (Round your answers to the nearest cent.) 30-year loan $ 37718.75 25-year loan $ (b) Calculate the savings in interest (in dollars) by using the 25-year loan. (Round your answer to the nearest cent.) $
1. **State the problem:**
We need to calculate the monthly payments for two loans of $355,000 each, one for 30 years and one for 25 years, both at an annual interest rate of 4.25%. Then, find the savings in interest by choosing the 25-year loan.
2. **Formula for monthly payment:**
The monthly payment $M$ for a loan is given by the formula:
$$
M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
$$
where:
- $P$ is the loan principal ($355,000$),
- $r$ is the monthly interest rate (annual rate divided by 12),
- $n$ is the total number of monthly payments (years $\times 12$).
3. **Calculate monthly interest rate:**
Annual interest rate = $4.25\% = 0.0425$
$$
r = \frac{0.0425}{12} = 0.0035416667
$$
4. **Calculate $n$ for each loan:**
- For 30 years: $n = 30 \times 12 = 360$
- For 25 years: $n = 25 \times 12 = 300$
5. **Calculate monthly payment for 30-year loan:**
$$
M_{30} = 355000 \times \frac{0.0035416667(1+0.0035416667)^{360}}{(1+0.0035416667)^{360} - 1}
$$
Calculate powers:
$$
(1+0.0035416667)^{360} = (1.0035416667)^{360} \approx 3.524
$$
Substitute:
$$
M_{30} = 355000 \times \frac{0.0035416667 \times 3.524}{3.524 - 1} = 355000 \times \frac{0.01248}{2.524} = 355000 \times 0.004945
$$
$$
M_{30} \approx 1755.48
$$
6. **Calculate monthly payment for 25-year loan:**
$$
M_{25} = 355000 \times \frac{0.0035416667(1+0.0035416667)^{300}}{(1+0.0035416667)^{300} - 1}
$$
Calculate powers:
$$
(1.0035416667)^{300} \approx 2.853
$$
Substitute:
$$
M_{25} = 355000 \times \frac{0.0035416667 \times 2.853}{2.853 - 1} = 355000 \times \frac{0.0101}{1.853} = 355000 \times 0.00545
$$
$$
M_{25} \approx 1933.05
$$
7. **Calculate total payments and interest for each loan:**
- Total paid for 30-year loan:
$$
1755.48 \times 360 = 631972.80
$$
- Total interest for 30-year loan:
$$
631972.80 - 355000 = 276972.80
$$
- Total paid for 25-year loan:
$$
1933.05 \times 300 = 579915.00
$$
- Total interest for 25-year loan:
$$
579915.00 - 355000 = 224915.00
$$
8. **Calculate savings in interest:**
$$
276972.80 - 224915.00 = 52057.80
$$
**Final answers:**
- Monthly payment for 30-year loan: $1755.48$
- Monthly payment for 25-year loan: $1933.05$
- Savings in interest by choosing 25-year loan: $52057.80$