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I'll go through each question one at a time and provide solutions. Let's start with Question 1.

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### 1. Solution for Question 1

#### Question 1:
*Suppose Mr. and Mrs. Mariano deposit ₱20,000.00 at the end of each year for 5 years in an investment account that earns 10% per year compounded annually. What is the amount of this annuity?*

This is a **future value of an ordinary annuity** problem, where payments are made at the end of each period.

#### Formula:
The future value of an ordinary annuity (FV) is given by:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
where:
- \( P = 20,000 \) (annual deposit)
- \( r = 0.10 \) (interest rate per period, compounded annually)
- \( n = 5 \) (number of years)

#### Solution:
\[
FV = 20,000 \times \frac{(1 + 0.10)^5 - 1}{0.10}
\]
Calculating each part:
1. \( (1 + 0.10)^5 = 1.10^5 \approx 1.61051 \)
2. \( 1.61051 - 1 = 0.61051 \)
3. \( \frac{0.61051}{0.10} = 6.1051 \)
4. \( 20,000 \times 6.1051 = 122,102 \)

**Future Value (Amount of the annuity) = ₱122,102**

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### Next Steps
Would you like me to continue with the other questions, or do you need more details for this one?

#### Additional Questions to Consider:
1. What if the deposit period was 10 years instead of 5 years?
2. How would the final amount change if the interest rate was 8% instead of 10%?
3. How would the formula adjust if payments were made at the beginning of each period (annuity due)?
4. What would happen if compounding was quarterly instead of annually?
5. How much would Mr. and Mrs. Mariano earn if they stopped depositing after 3 years?

**Tip:** For annuity problems, always check if the payment is at the beginning or end of the period, as this affects the type of annuity formula you use.

1. Suppose Mr. and Mrs. Mariano deposit ₱20,000.00 at the end of each year for 5 years in an investment account that earns 10% per year compounded annually. What is the amount of this annuity? 2. If you pay ₱500.00 at the end of each month for 40 years in an account that pays interest at 10% compounded monthly, how much money do you have after 40 years? 3. Aaron's mother saved ₱5,000 at the end of every 6 months in an educational plan that earns 6% per year compounded semi-annually. What is the amount at the end of 18 years? How much interest is earned? 4. Rose wants to have enough money in her retirement account when she reaches the age of 60. She plans to withdraw ₱36,000 every 3 months for 20 years starting 3 months after she retires. How much should she deposit at retirement at 12% per year compounded quarterly for the annuity? 5. Fred borrowed money to buy a motorcycle. He will repay the loan by making monthly payments of ₱1,500 per month for the next 24 months at an interest rate of 9% per year compounded monthly. How much did Fred borrow? How much interest does Fred pay?

Learn how to solve annuity and loan problems involving compound interest. Includes examples such as calculating the future value of a retirement account, determining monthly savings for long-term growth, and solving loan repayment schedules with detailed step-by-step solutions. Suitable for students in Grades 10-12.

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