To solve these word problems, we will use the formula for the future value or present value of annuities, along with the concept of compound interest. Let’s go through each problem one by one.

Problem 1

Gina pays P21,500 at the end of each six months for 10 years, and the money is worth 3% compounded monthly.

This is an ordinary annuity problem (since payments are made at the end of each period) where we need to find the present value.

1. Payment (PMT) = P21,500
2. Interest Rate (i) = 3% compounded monthly. Convert to effective six-monthly rate:
   • Monthly interest rate $= \frac{3\%}{12} = 0.25\%$
   • Six-month interest rate $= (1 + 0.0025)^{6} - 1 = 0.0151$ or 1.51%
3. Number of Periods (n) = 10 years × 2 periods per year = 20 periods

The present value of an ordinary annuity is given by: $PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}$

Plugging in the values: $PV = 21500 \times \frac{1 - (1 + 0.0151)^{-20}}{0.0151}$

Calculating this gives: $PV \approx P377,185.13$

So, the cost of the car is approximately P377,185.13.

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Problem 2

P7,000 will be invested every two years in a bank at 4% compounded semi-annually. Find the size of the fund at the end of eight years.

This is a future value annuity problem where the annuity payments are made every two years.

1. Payment (PMT) = P7,000
2. Interest Rate (i) = 4% compounded semi-annually. Effective rate every two years:
   • Semi-annual rate $= \frac{4\%}{2} = 2\%$
   • Two-year effective rate $= (1 + 0.02)^4 - 1 = 0.0824$ or 8.24%
3. Number of Periods (n) = 8 years / 2 years per period = 4 periods

The future value of an annuity is given by: $FV = PMT \times \frac{(1 + i)^n - 1}{i}$

Plugging in the values: $FV = 7000 \times \frac{(1 + 0.0824)^4 - 1}{0.0824}$

Calculating this gives: $FV \approx P31,341.85$

So, the size of the fund at the end of eight years is approximately P31,341.85.

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Problem 3

Bea agrees to pay P400 at the beginning of every three months for five years. The money is worth 5% compounded monthly.

This is an annuity due problem (since payments are made at the beginning of each period) where we need to find the present value.

1. Payment (PMT) = P400
2. Interest Rate (i) = 5% compounded monthly. Effective three-monthly rate:
   • Monthly interest rate $= \frac{5\%}{12} \approx 0.4167\%$
   • Three-monthly interest rate $= (1 + 0.004167)^3 - 1 = 0.0125$ or 1.25%
3. Number of Periods (n) = 5 years × 4 periods per year = 20 periods

The present value of an annuity due is given by: $PV = PMT \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i)$

Plugging in the values: $PV = 400 \times \frac{1 - (1 + 0.0125)^{-20}}{0.0125} \times (1 + 0.0125)$

Calculating this gives: $PV \approx P7,327.90$

So, Bea borrowed approximately P7,327.90.

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If you need further details on any of these solutions or have questions, feel free to ask!

Here are some related questions for deeper understanding:

1. What is the difference between annuity due and ordinary annuity?
2. How does the compounding frequency affect the effective interest rate?
3. What is the impact of payment frequency on the present and future value of annuities?
4. Why is it necessary to adjust the interest rate to match the payment periods in annuity calculations?
5. How can annuity formulas be applied in real-life financial decisions?

Tip: When working with annuities, always ensure the interest rate is adjusted to the same period as the payments (e.g., annual, semi-annual) to get accurate results.