Responder
To pay off a \$9,000 credit card balance with a 12.5% APR over 3 years, you'll need to make monthly payments of approximately \$301.08 each. Over 36 months, the total amount you'll pay is \$10,838.88.
Solución
To solve this problem, we need to determine the monthly payment required to pay off a credit card balance of \( \$9000 \) over 3 years with an annual percentage rate (APR) of \( 12.5\% \).
### Step 1: Convert APR to Monthly Interest Rate
The monthly interest rate \( r \) can be calculated from the APR as follows:
\[
r = \frac{\text{APR}}{12} = \frac{12.5\%}{12} = \frac{0.125}{12}
\]
### Step 2: Calculate the Number of Payments
The total number of monthly payments \( n \) over 3 years is:
\[
n = 3 \times 12 = 36
\]
### Step 3: Use the Formula for Monthly Payment
The formula for the monthly payment \( M \) on an amortizing loan is given by:
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
where:
- \( P \) is the principal amount (the initial balance),
- \( r \) is the monthly interest rate,
- \( n \) is the number of payments.
### Step 4: Calculate Monthly Payment
Now we will calculate \( M \) using the values we have:
- \( P = 9000 \)
- \( r = \frac{0.125}{12} \)
- \( n = 36 \)
Let's calculate \( M \).
First, we need to calculate \( r \):
\[
r = \frac{0.125}{12} \approx 0.01041667
\]
Now we can substitute these values into the formula for \( M \):
\[
M = 9000 \frac{0.01041667(1 + 0.01041667)^{36}}{(1 + 0.01041667)^{36} - 1}
\]
Now, I will perform the calculation for \( M \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{9000\left(0.01041667\left(1+0.01041667\right)^{36}\right)}{\left(\left(1+0.01041667\right)^{36}-1\right)}\)
- step1: Remove the parentheses:
\(\frac{9000\times 0.01041667\left(1+0.01041667\right)^{36}}{\left(1+0.01041667\right)^{36}-1}\)
- step2: Add the numbers:
\(\frac{9000\times 0.01041667\times 1.01041667^{36}}{\left(1+0.01041667\right)^{36}-1}\)
- step3: Add the numbers:
\(\frac{9000\times 0.01041667\times 1.01041667^{36}}{1.01041667^{36}-1}\)
- step4: Convert the expressions:
\(\frac{9000\times 0.01041667\left(\frac{101041667}{100000000}\right)^{36}}{1.01041667^{36}-1}\)
- step5: Convert the expressions:
\(\frac{9000\times 0.01041667\left(\frac{101041667}{100000000}\right)^{36}}{\left(\frac{101041667}{100000000}\right)^{36}-1}\)
- step6: Multiply:
\(\frac{\frac{9375003\times 101041667^{36}}{10^{293}}}{\left(\frac{101041667}{100000000}\right)^{36}-1}\)
- step7: Subtract the numbers:
\(\frac{\frac{9375003\times 101041667^{36}}{10^{293}}}{\frac{101041667^{36}-100000000^{36}}{100000000^{36}}}\)
- step8: Multiply by the reciprocal:
\(\frac{9375003\times 101041667^{36}}{10^{293}}\times \frac{100000000^{36}}{101041667^{36}-100000000^{36}}\)
- step9: Rewrite the expression:
\(\frac{9375003\times 101041667^{36}}{10^{293}}\times \frac{10^{288}}{101041667^{36}-100000000^{36}}\)
- step10: Reduce the numbers:
\(\frac{9375003\times 101041667^{36}}{10^{5}}\times \frac{1}{101041667^{36}-100000000^{36}}\)
- step11: Multiply the fractions:
\(\frac{9375003\times 101041667^{36}}{10^{5}\left(101041667^{36}-100000000^{36}\right)}\)
- step12: Multiply:
\(\frac{9375003\times 101041667^{36}}{10^{5}\times 101041667^{36}-10^{293}}\)
The monthly payment \( M \) is approximately \( 301.08 \).
### Step 5: Calculate Total Amount Paid
To find the total amount paid over the 3 years, we multiply the monthly payment by the total number of payments:
\[
\text{Total Amount Paid} = M \times n = 301.08 \times 36
\]
Now, let's calculate the total amount paid.
Calculate the value by following steps:
- step0: Calculate:
\(301.08\times 36\)
- step1: Multiply the numbers:
\(10838.88\)
The total amount paid over the 3 years is approximately \( 10838.88 \).
### Final Answer
Thus, the total amount you will pay to pay off the credit card balance in 3 years is **$10,838.88**.
Revisado y aprobado por el equipo de tutoría de UpStudy
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