What is the payment on a $250,000 mortgage?
Understand the Problem
The question is asking for the monthly payment amount for a $250,000 mortgage. To solve this, we typically need to know the interest rate and the term of the mortgage (e.g., 30 years), as these factors significantly influence the payment amount.
Answer
$1207.05$
Answer for screen readers
The monthly payment for the $250,000 mortgage is approximately $1207.05.
Steps to Solve
- Identify the mortgage variables
For a mortgage, we need the principal amount ($P$), interest rate ($r$), and the number of payments ($n$). In this case:
- Principal ($P$) = $250,000
- Annual interest rate = (assumed) 4% or 0.04
- Mortgage term = 30 years
- Convert annual interest rate to monthly
To calculate the monthly interest rate, divide the annual interest rate by 12:
$$ r = \frac{0.04}{12} = 0.0033333 $$
- Calculate the total number of payments
The total number of monthly payments for a 30-year mortgage is:
$$ n = 30 \times 12 = 360 $$
- Use the mortgage payment formula
The formula to calculate the monthly payment ($M$) is:
$$ M = P \frac{r(1+r)^n}{(1+r)^n - 1} $$
Substituting the values we have:
$$ M = 250000 \frac{0.0033333(1+0.0033333)^{360}}{(1+0.0033333)^{360}-1} $$
- Calculate the payment amount
First, calculate $(1 + r)^n$:
$$ (1 + 0.0033333)^{360} \approx 3.2434 $$
Next, calculate the payment:
$$ M = 250000 \frac{0.0033333 \times 3.2434}{3.2434 - 1} $$
Simplify this to find $M$.
$$ M = 250000 \frac{0.0108113}{2.2434} $$
$$ M \approx 250000 \times 0.0048282 \approx 1207.05 $$
The monthly payment for the $250,000 mortgage is approximately $1207.05.
More Information
The monthly payment can vary based on the interest rate and loan term. A lower interest rate or shorter loan term generally results in a higher monthly payment, while a longer loan term spreads the payments over more months, resulting in smaller payments.
Tips
- Not converting the annual interest rate to a monthly rate; ensure to divide by 12.
- Forgetting to account for the total number of payments over the full term of the mortgage (360 payments for a 30-year mortgage).
- Incorrectly applying the mortgage payment formula, especially in the calculations of $r$ and $n$.
