Question

Loan program with 0% on loans repaid within 5 yrs. Person took out loan from this company and made equal payments
After 6 months loan balance was 12,150
After 11 months loan balance was 11,025
Which function can be used to find y the balance of loan after x monthly payments

Answer 1

To find a function that represents the balance of the loan \(y\) after \(x\) monthly payments, we can use the information provided.

Let's denote:
- \(B_0\) as the initial loan balance,
- \(p\) as the amount of each monthly payment,
- \(r\) as the monthly interest rate (which is 0% in this case),
- \(B(x)\) as the balance after \(x\) months.

1. **Understanding the balances given:**
- After 6 months, the balance is \(12,150\).
- After 11 months, the balance is \(11,025\).

2. **Writing equations:**
Since the loan has zero interest, the loan balance decreases linearly with each monthly payment. We can express the balance after \(x\) months as:
\[
B(x) = B_0 - p \cdot x
\]

3. **Substituting the known values:**
We have two equations based on the monthly payments.

Using the balance after 6 months:
\[
B(6) = B_0 - 6p = 12,150 \quad \text{(1)}
\]

Using the balance after 11 months:
\[
B(11) = B_0 - 11p = 11,025 \quad \text{(2)}
\]

4. **Subtracting equations (1) from (2)**:
Now, we can eliminate \(B_0\) by subtracting the first equation from the second.
\[
(B_0 - 11p) - (B_0 - 6p) = 11,025 - 12,150
\]
This simplifies to:
\[
-5p = -1,125 \Rightarrow p = \frac{1,125}{5} = 225
\]

5. **Finding \(B_0\)**:
Substitute \(p\) back into one of the previous equations to find \(B_0\). Using equation (1):
\[
B_0 - 6(225) = 12,150
\]
\[
B_0 - 1,350 = 12,150 \Rightarrow B_0 = 12,150 + 1,350 = 13,500
\]

6. **Final function**:
Now we have \(B_0 = 13,500\) and \(p = 225\). The function for the balance after \(x\) months is:
\[
B(x) = 13,500 - 225x
\]

In conclusion, the function that represents the balance of the loan after \(x\) monthly payments is:
\[
y = 13,500 - 225x
\]