Principal $14,000, Interest Rate 7%, Time 240 days (use ordinary interest)

Partial payments: On 100th day, $5,400; On 180th day, $3,200, what is the total intrest cost, balance after the payment on the 100th day and 180th day, and the final payment

To solve this problem, we will use the **ordinary interest method** (where a year is assumed to have 360 days). The interest is calculated using the formula:

\[
I = P \times r \times \frac{t}{360}
\]

Where:
- $ P $ is the principal amount.
- $ r $ is the interest rate as a decimal (7% = 0.07).
- $ t $ is the time in days.

We will break this into several steps:

1. **Interest calculation up to the 100th day**.
2. **Balance after the $5,400 payment on the 100th day**.
3. **Interest calculation from the 100th day to the 180th day**.
4. **Balance after the $3,200 payment on the 180th day**.
5. **Interest calculation from the 180th day to the 240th day (final payment)**.
6. **Total interest cost**.

### Step 1: Interest Calculation up to the 100th Day

- Principal: $ P = 14,000 $
- Interest rate: $ r = 7\% = 0.07 $
- Time: $ t = 100 $ days

\[
I_1 = 14,000 \times 0.07 \times \frac{100}{360}
\]

### Step 2: Balance After $5,400 Payment on the 100th Day

After finding the interest accrued on the 100th day, subtract the payment of $5,400 from the sum of principal and interest. The new balance will be calculated.

### Step 3: Interest Calculation from the 100th Day to the 180th Day

Using the new principal (after payment) and calculating interest for the next 80 days (from day 100 to day 180):

\[
I_2 = \text{New principal} \times 0.07 \times \frac{80}{360}
\]

### Step 4: Balance After $3,200 Payment on the 180th Day

After finding the interest accrued between day 100 and 180, subtract the $3,200 payment from the new principal and interest.

### Step 5: Interest Calculation from the 180th Day to the 240th Day

Finally, calculate the interest for the remaining 60 days (from day 180 to day 240) on the updated principal.

\[
I_3 = \text{New principal} \times 0.07 \times \frac{60}{360}
\]

### Step 6: Final Payment and Total Interest Cost

The final payment will be the sum of the principal balance after the 180th day plus the interest accrued in the last 60 days. The total interest cost will be the sum of all interest accrued during the three periods.

Let's calculate the values step by step.Here are the results for each step:

1. **Interest up to the 100th day**: 
   \[
   I_1 = \$272.22
   \]

2. **Balance after the $5,400 payment on the 100th day**: 
   \[
   \text{New balance} = \$8,872.22
   \]

3. **Interest from the 100th day to the 180th day**: 
   \[
   I_2 = \$138.01
   \]

4. **Balance after the $3,200 payment on the 180th day**: 
   \[
   \text{New balance} = \$5,810.23
   \]

5. **Interest from the 180th day to the 240th day**: 
   \[
   I_3 = \$67.79
   \]

6. **Final payment on the 240th day**: 
   \[
   \text{Final payment} = \$5,878.02
   \]

7. **Total interest cost**: 
   \[
   \text{Total interest} = \$478.02
   \]

Let me know if you'd like more details or have any questions!

### Related Questions:
1. How would the interest calculation differ if compound interest were used instead of ordinary interest?
2. What would be the final payment if no partial payments were made during the 240 days?
3. How would the interest cost change if the interest rate was 5% instead of 7%?
4. What is the impact of making a payment later, such as on the 150th day instead of the 100th?
5. How would the calculation change if using exact interest (365 days per year) instead of ordinary interest?

### Tip:
Paying off loans earlier in the period can significantly reduce the overall interest cost, as less principal is exposed to interest accumulation over time.

Principal $14,000, Interest Rate 7%, Time 240 days (use ordinary interest). Partial payments: On 100th day, $5,400; On 180th day, $3,200. What is the total interest cost, balance after the payment on the 100th day and 180th day, and the final payment?

Learn how to calculate simple interest and balance for a $14,000 loan at 7% interest over 240 days, with partial payments on the 100th and 180th days. Understand step-by-step how to calculate the interest, balance after each payment, and the final payment amount.

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