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answer the first pictures and make the second picture as guide or reference

The first image shows a problem related to compound interest, while the second image provides a solution framework. Let's break down the problem and solution.

### Problem Description
- **Principal (P)**: ₱10,000
- **Interest rate (r)**: 12% per annum, compounded monthly
- **Time periods**:
  1. 1 year
  2. 4 months
  3. 2 years

### Reference Solution Approach
The second image outlines the formula used for compound interest:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (₱10,000).
- \( r \) is the annual interest rate (12% or 0.12).
- \( n \) is the number of times that interest is compounded per year (monthly compounding, so \( n = 12 \)).
- \( t \) is the time in years.

### Solutions

1. **For 1 year:**
   \[
   A = 10,000 \left(1 + \frac{0.12}{12}\right)^{12 \times 1}
   \]
   \[
   A = 10,000 \left(1 + 0.01\right)^{12}
   \]
   \[
   A = 10,000 \left(1.01\right)^{12} \approx 10,000 \times 1.1268 \approx 11,268 \text{ (₱)}
   \]

2. **For 4 months:**
   \[
   A = 10,000 \left(1 + \frac{0.12}{12}\right)^{12 \times \frac{4}{12}}
   \]
   \[
   A = 10,000 \left(1 + 0.01\right)^{4}
   \]
   \[
   A = 10,000 \times (1.01)^4 \approx 10,000 \times 1.0406 \approx 10,406 \text{ (₱)}
   \]

3. **For 2 years:**
   \[
   A = 10,000 \left(1 + \frac{0.12}{12}\right)^{12 \times 2}
   \]
   \[
   A = 10,000 \left(1 + 0.01\right)^{24}
   \]
   \[
   A = 10,000 \times (1.01)^{24} \approx 10,000 \times 1.2682 \approx 12,682 \text{ (₱)}
   \]

### Summary of Results
- **1 year:** ₱11,268
- **4 months:** ₱10,406
- **2 years:** ₱12,682

### Would you like more details on any of the steps? Here are some additional questions you might find useful:

1. How does the compounding frequency affect the total amount of interest earned?
2. What would be the difference if the interest was compounded annually instead of monthly?
3. How would the formula change if interest were compounded daily instead of monthly?
4. What happens to the total amount if the interest rate increases to 15% while keeping everything else constant?
5. How would you calculate the time required for the principal to double using the given interest rate?

**Tip:** When dealing with compound interest, the frequency of compounding has a significant effect on the final amount. The more frequently the interest is compounded, the higher the final amount will be.

Solve the compound interest problem where the principal is ₱10,000, the interest rate is 12% per annum compounded monthly, and the time periods are 1 year, 4 months, and 2 years.

Learn how to calculate compound interest on a principal amount of ₱10,000 with an annual interest rate of 12% compounded monthly. This example covers time periods of 1 year, 4 months, and 2 years, using the compound interest formula.

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