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define if it's compound interest or annuities then solve it,show the full solution 

The problems provided involve **annuities**, as they refer to regular payments made over time. Specifically:

1. **Kenyo's Loan**: This is a **present value of an ordinary annuity problem**. Kenyo is making regular payments of Php 3,000 every 3 months to repay a loan with interest compounded quarterly.

2. **Sasa's Savings**: This is a **future value of an annuity problem**. Sasa is regularly saving Php 5,000 every 6 months, and the savings grow with interest compounded monthly.

### Problem 1: Kenyo's Loan

We need to calculate the present value of an annuity (P), which represents how much Kenyo borrowed. The formula for the present value of an ordinary annuity is:

\[
P = A \times \left( \frac{1 - (1 + i)^{-n}}{i} \right)
\]

Where:
- \( A = 3,000 \) (the regular payment)
- \( i = \frac{8\%}{4} = 0.02 \) (the quarterly interest rate, since interest is compounded quarterly)
- \( n = 4 \times 4 = 16 \) (the total number of quarters over 4 years)

#### Step-by-Step Calculation:

1. \( A = 3,000 \)
2. \( i = 0.02 \)
3. \( n = 16 \)

Substitute into the formula:

\[
P = 3,000 \times \left( \frac{1 - (1 + 0.02)^{-16}}{0.02} \right)
\]

Calculate:

\[
1 + 0.02 = 1.02
\]
\[
1.02^{-16} \approx 0.7260
\]
\[
1 - 0.7260 = 0.2740
\]
\[
\frac{0.2740}{0.02} = 13.700
\]
\[
P = 3,000 \times 13.700 = 41,100
\]

**Kenyo borrowed Php 41,100**.

### Problem 2: Sasa's Savings

This is a future value of an annuity problem. The formula for the future value of an annuity is:

\[
F = A \times \left( \frac{(1 + i)^n - 1}{i} \right)
\]

Where:
- \( A = 5,000 \) (the regular savings deposit)
- \( i = \frac{0.25\%}{12} = 0.0020833 \) (monthly interest rate, since interest is compounded monthly)
- \( n = 10 \times 2 = 20 \) (number of deposits in 10 years, since Sasa deposits every 6 months)

#### Step-by-Step Calculation:

1. \( A = 5,000 \)
2. \( i = 0.0020833 \)
3. \( n = 20 \)

Substitute into the formula:

\[
F = 5,000 \times \left( \frac{(1 + 0.0020833)^{120} - 1}{0.0020833} \right)
\]

Calculate:

\[
1 + 0.0020833 = 1.0020833
\]
\[
1.0020833^{120} \approx 1.283
\]
\[
1.283 - 1 = 0.283
\]
\[
\frac{0.283}{0.0020833} = 135.91
\]
\[
F = 5,000 \times 135.91 = 679,550
\]

**Sasa's savings after 10 years will be Php 679,550**.

---

Would you like more details or clarifications?

### Related Questions:
1. What is the difference between ordinary annuity and annuity due?
2. How does the frequency of compounding affect the future value of an investment?
3. What happens if Kenyo pays more or less than Php 3,000 every 3 months?
4. How would a change in the interest rate affect Sasa's final savings?
5. Can you apply annuity formulas to scenarios other than loans and savings?

### Tip:
Always double-check the interest compounding frequency when dealing with annuities or compound interest problems—it significantly affects the calculations.

Kenyo is paying Php 3,000 every 3 months for the amount he borrowed at an interest rate of 8% compounded quarterly. How much did he borrow if he agreed that the loan will be paid in 4 years? Sasa, an accountant, saves Php 5,000 every 6 months in a cooperative that pays 0.25% compounded monthly. How much will be her savings after 10 years?

This task covers how to calculate the present value of Kenyo's loan using an annuity formula with a 4-year term and 8% compounded quarterly interest. It also calculates the future value of Sasa's 10-year savings plan with a 0.25% interest rate compounded monthly.

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