A loan of ​$1050 is being paid off by quarterly payments of ​$120. If interest is at the rate of 14​% compounded​ quarterly, how many full payments will be​ made?

To solve this problem, we can use the **loan amortization formula**, which takes into account periodic payments on a loan with compound interest. Here’s the breakdown of how we’ll approach it:

1. **Given Information:**
   - Principal loan amount ($P$) = $1050
   - Quarterly payment ($R$) = $120
   - Interest rate = 14% compounded quarterly, which means the quarterly interest rate is:
     \[
     i = \frac{14\%}{4} = 0.035 \text{ or } 3.5\%
     \]
2. **Amortization Formula:**
   The formula to calculate the number of payments ($n$) required to pay off a loan is:
   \[
   n = \frac{\ln\left(\frac{R}{R - Pi}\right)}{\ln(1 + i)}
   \]
   where:
   - $R$ is the periodic payment,
   - $P$ is the initial loan principal,
   - $i$ is the interest rate per period.

3. **Substitute the Values:**
   Plugging in the values:
   \[
   n = \frac{\ln\left(\frac{120}{120 - 1050 \times 0.035}\right)}{\ln(1 + 0.035)}
   \]

Let's calculate this step-by-step.

### Step-by-Step Calculation:
1. Calculate $Pi = 1050 \times 0.035 = 36.75$.
2. Calculate $R - Pi = 120 - 36.75 = 83.25$.
3. Calculate $\frac{R}{R - Pi} = \frac{120}{83.25} \approx 1.4417$.
4. Find $\ln(1.4417) \approx 0.3653$.
5. Find $\ln(1 + i) = \ln(1.035) \approx 0.0344$.
6. Now calculate $n$:
   \[
   n = \frac{0.3653}{0.0344} \approx 10.61
   \]

Since only **full payments** are counted, round up to 11 payments. 

### Answer:
The number of full payments needed to pay off the loan is **11 payments**.

Would you like details or have any questions?

---

1. What if the loan had monthly payments instead of quarterly?
2. How would the number of payments change with a lower interest rate?
3. Can you explain how the compound interest affects the number of payments?
4. What happens to the total interest paid if the loan is paid off faster?
5. How can one calculate the final amount if extra payments are made?

**Tip:** Always round up to the next whole number when calculating loan payments, as partial payments are generally not allowed.

A loan of $1050 is being paid off by quarterly payments of $120. If interest is at the rate of 14% compounded quarterly, how many full payments will be made?

Discover how to calculate the number of payments required to pay off a $1050 loan with quarterly payments of $120 and a 14% interest rate compounded quarterly. This step-by-step solution explains the loan amortization formula, interest calculations, and logarithmic applications. Ideal for Grades 11-12.

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