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the formula for compound interest is $$ A=P\left(1+r\right)^{t} $$$$  $$

The problem involves calculating compound interest, where Alex deposits ₱1,000.00 in a bank with an annual interest rate of 5%. Let's break down and solve both parts.

### Compound Interest Formula:
The general formula for compound interest is:
\[
A = P \left(1 + r \right)^t
\]
Where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (₱1,000.00 in this case).
- $ r $ is the annual interest rate (5%, which is 0.05 as a decimal).
- $ t $ is the time in years.

### Part a: Equation for the Investment
Using the formula for compound interest, the equation that best describes the investment after $ t $ years is:
\[
A = 1{,}000 \left(1 + 0.05\right)^t
\]
Simplified:
\[
A = 1{,}000 \left(1.05\right)^t
\]
This is the equation for the investment over $ t $ years.

### Part b: How much money will he have after 2 years?
For $ t = 2 $, we substitute into the equation:
\[
A = 1{,}000 \left(1.05\right)^2
\]
Calculating:
\[
A = 1{,}000 \times 1.1025 = 1{,}102.50
\]
So, after 2 years, Alex will have ₱1,102.50.

Would you like further details or clarifications on any part of this solution?

### Follow-up Questions:
1. What happens to the amount if the interest rate increases to 6%?
2. How would the formula change if the interest was compounded monthly instead of annually?
3. What is the accumulated amount after 5 years at 5% interest?
4. How much interest would Alex earn in 2 years?
5. If the interest was compounded quarterly, how much would he have after 2 years?

### Tip:
Always ensure you convert the interest rate into a decimal (e.g., 5% becomes 0.05) before applying it in the formula.

Alex deposited ₱1,000.00 in a bank at Puerto Galera that pays 5% compound interest annually. a. What equation best describes this investment after t years? b. How much money will he have after 2 years?

Explore how to calculate compound interest using the formula A = P(1 + r)^t. In this example, Alex deposits ₱1,000 in a bank at Puerto Galera with 5% annual interest. Find the equation for the investment and determine how much Alex will have after 2 years.

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