From the above formulas which is the formula for the payment M to be given 12 times a year for mortgage on a principal P, at annual rate r, in order to have paid the loan off in ti... From the above formulas which is the formula for the payment M to be given 12 times a year for mortgage on a principal P, at annual rate r, in order to have paid the loan off in time t (measured in years)?
Understand the Problem
The question is asking which of the provided formulas corresponds to the payment amount (M) that needs to be made 12 times a year for a mortgage on a principal amount (P) with a specified annual interest rate (r), in order to pay off the loan in a given time period (t) measured in years.
Answer
I choose (f).
Answer for screen readers
I choose (f).
Steps to Solve
- Identify the Payment Formula for Mortgages
The goal is to find the formula for the monthly payment $M$ for a loan with principal $P$, an annual interest rate $r$, paid off over $t$ years.
- Understand Variables for Monthly Payments
A monthly payment is calculated based on the monthly interest rate and the total number of payments.
- Monthly interest rate: $r_m = \frac{r}{12}$
- Total number of payments: $n = 12t$
- Select the Correct Formula from Options
From the given formulas, we compare them to the standard mortgage payment formula:
$$ M = P \cdot \frac{r_m(1 + r_m)^n}{(1 + r_m)^n - 1} $$
- Relate the Formula to Options
Checking option (f):
$$ M = - \frac{P \cdot \frac{r}{12}}{1 - (1 + \frac{r}{12})^{-12t}} $$
This resembles the common formula for mortgage payments, confirming that option (f) accounts for the monthly pay.
I choose (f).
More Information
The formula for monthly payments on a mortgage accounts for both the principal and the monthly interest, making it essential for budgeting monthly mortgage payments. Understanding the components of the formula helps in financial planning.
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