If you borrow money at a 7% APR and make $600 monthly payments over the course of 10 years to pay back the loan, then how much money did you borrow?

To determine how much money was borrowed, we can use the formula for the present value of an annuity, since you're making fixed monthly payments over a period of time.

The formula for the present value $ PV $ of an annuity is:

\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- $ P $ is the monthly payment ($600),
- $ r $ is the monthly interest rate (which is the annual percentage rate divided by 12),
- $ n $ is the total number of payments (which is the number of years multiplied by 12).

### Step 1: Calculate the monthly interest rate
The annual percentage rate (APR) is 7%, so the monthly interest rate is:

\[
r = \frac{7\%}{12} = \frac{0.07}{12} = 0.0058333
\]

### Step 2: Calculate the total number of payments
Since the loan is to be paid over 10 years with monthly payments, the total number of payments is:

\[
n = 10 \times 12 = 120 \text{ payments}
\]

### Step 3: Use the formula
Now, we can plug in the values:

\[
PV = 600 \times \frac{1 - (1 + 0.0058333)^{-120}}{0.0058333}
\]

Let’s compute that:

\[
PV = 600 \times \frac{1 - (1.0058333)^{-120}}{0.0058333}
\]

First, calculate $ (1.0058333)^{-120} $:

\[
(1.0058333)^{-120} \approx 0.49674
\]

Now compute the rest:

\[
PV = 600 \times \frac{1 - 0.49674}{0.0058333} = 600 \times \frac{0.50326}{0.0058333}
\]

\[
PV \approx 600 \times 86.3168 = 51,790.08
\]

### Conclusion:
The amount of money you originally borrowed was approximately **$51,790.08**.

Would you like further details or have any questions?

### Related Questions:
1. How does changing the interest rate affect the loan amount?
2. What if the payments were bi-weekly instead of monthly?
3. How would the loan amount change if the term was extended to 15 years?
4. Can you explain how APR differs from APY (Annual Percentage Yield)?
5. How much total interest is paid over the course of the loan?

### Tip:
When dealing with loans, always compare the APR across different offers, as small differences in interest rates can significantly affect the total repayment amount.

Learn how to calculate the loan amount borrowed at a 7% APR with $600 monthly payments over 10 years using the present value of an annuity formula. This detailed solution breaks down interest rates, monthly payments, and the number of periods involved. Perfect for high school seniors or college students.

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