15. Using exact number of days, the money borrowed earns P120.00 at 12% rate. What is the accumulated amount from January 12, 2011 to September 24, 2012?

To find the accumulated amount, we first need to determine the principal amount that was borrowed, which earned ₱120 in interest at a 12% annual interest rate over the specified period. Then, we can calculate the accumulated amount by adding the interest earned to the principal.

### Step 1: Calculate the Number of Days

The period from **January 12, 2011** to **September 24, 2012** spans two years:

1. **From January 12, 2011, to January 12, 2012:** 1 year = 365 days
2. **From January 12, 2012, to September 24, 2012:**
   - January 12 to January 31: 19 days
   - February: 29 days (2012 is a leap year)
   - March: 31 days
   - April: 30 days
   - May: 31 days
   - June: 30 days
   - July: 31 days
   - August: 31 days
   - September 1 to September 24: 24 days

Total days:
\[
t = 365 + 19 + 29 + 31 + 30 + 31 + 30 + 31 + 31 + 24 = 621 \text{ days}
\]

### Step 2: Determine the Principal Amount

Given:
- Interest earned (\( I \)) = ₱120.00
- Interest rate (\( r \)) = 12% (or 0.12)
- Time (\( t \)) = 621 days

The formula for simple interest is:
\[
I = \frac{P \times r \times t}{365}
\]
Rearrange to solve for \( P \):
\[
P = \frac{I \times 365}{r \times t}
\]
Substitute the values:
\[
P = \frac{120 \times 365}{0.12 \times 621} = \frac{43,800}{74.52} = 588.05 \text{ (rounded to the nearest centavo)}
\]

### Step 3: Calculate the Accumulated Amount

The accumulated amount \( A \) is given by:
\[
A = P + I
\]
Substitute the values:
\[
A = 588.05 + 120.00 = 708.05
\]

### Final Result:

The accumulated amount from January 12, 2011, to September 24, 2012, is **₱708.05**.

Would you like to explore further details or have additional questions?

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**Related Questions:**
1. How does the length of the borrowing period impact the accumulated amount?
2. What would the accumulated amount be with a different interest rate?
3. How does compound interest differ from simple interest in calculating accumulated amounts?
4. How do different borrowing terms (like monthly payments) affect the total cost?
5. What is the impact of rounding on financial calculations?

**Tip:** Knowing how to calculate the principal and accumulated amounts is key to understanding the true cost of borrowing and the benefits of investing over time.

Learn how to calculate the accumulated amount from January 12, 2011 to September 24, 2012, using simple interest. This problem involves determining the principal amount borrowed and calculating interest earned over 621 days at a 12% annual interest rate. Suitable for high school students.

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