Math Problem Statement

Compare the monthly payments and total loan costs for the following pairs of loan options. Assume that both loans are fixed rate and have the same closing costs. You need a ​$90 comma 000 loan. Option​ 1: a​ 30-year loan at an APR of 7.65​%. Option​ 2: a​ 15-year loan at an APR of 7.25​%. Question content area bottom Part 1 Find the monthly payment for each option. The monthly payment for option 1 is ​$    enter your response here. The monthly payment for option 2 is ​$    enter your response here. ​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

Solution

To find the monthly payment for each loan option, we can use the following formula for a fixed-rate mortgage:

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

Where:

  • MM is the monthly payment,
  • PP is the loan amount,
  • rr is the monthly interest rate (annual interest rate divided by 12),
  • nn is the total number of payments (loan term in months).

Option 1: 30-year loan at 7.65% APR

  1. Loan amount P=90,000P = 90,000
  2. Annual interest rate rannual=7.65%=0.0765r_{\text{annual}} = 7.65\% = 0.0765
    • Monthly

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Math Problem Analysis

Mathematical Concepts

Algebra
Finance Mathematics
Interest Calculation

Formulas

Monthly Payment Formula for Fixed-Rate Mortgages: M = P * [r(1+r)^n / (1+r)^n - 1]

Theorems

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Suitable Grade Level

Grades 10-12

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