A man wishes to bequeath to his daughter P20,000 ten years from now. What amount should he invest now if it will earn interest of 8% compounded annually during the first 5 years and 12% compounded quarterly during the next 5 years?

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To calculate the amount that the man needs to invest now, we can break down the problem into two parts: the first five years with an 8% interest rate compounded annually, and the next five years with a 12% interest rate compounded quarterly.

Step 1: Calculate the future value of P20,000 after five years with an 8% interest rate compounded annually.
Using the future value formula for compound interest:
FV = PV * (1 + r/n)^(n*t)

Where:
FV = Future Value
PV = Present Value (amount to be invested)
r = Annual interest rate (8%)
n = Number of times interest is compounded per year (annually)
t = Number of years (5 years)

Plugging in the values:
P20,000 = PV * (1 + 0.08/1)^(1*5)
20,000 = PV * (1.08)^5

To find PV, divide both sides by (1.08)^5:
PV = 20,000 / (1.08)^5

Step 2: Calculate the future value of PV after the next five years with a 12% interest rate compounded quarterly.
Using the future value formula for compound interest:
FV = PV * (1 + r/n)^(n*t)

Where:
FV = Future Value (P20,000)
PV = Present Value (result from Step 1)
r = Annual interest rate (12%)
n = Number of times interest is compounded per year (quarterly, so n = 4)
t = Number of years (5 years)

Plugging in the values:
20,000 = PV * (1 + 0.12/4)^(4*5)
20,000 = PV * (1.03)^20

To find PV, divide both sides by (1.03)^20:
PV = 20,000 / (1.03)^20

Step 3: Calculate the total amount that should be invested now.
To get the total amount to be invested now, add the present values obtained from Step 1 and Step 2:
Total investment = PV from Step 1 + PV from Step 2

Total investment = 20,000 / (1.08)^5 + 20,000 / (1.03)^20

Calculating this amount will provide you with the answer.

To calculate the present value that the man should invest now, we can use the formula for the future value of an investment compounded annually:

FV = PV(1 + r/n)^(n*t)

Where:
FV = future value (P20,000)
PV = present value (unknown)
r = interest rate (8% for the first 5 years, 12% for the next 5 years)
n = number of compounding periods per year (1 for the first 5 years, 4 for the next 5 years)
t = time in years (10 years)

Now, let's break the problem into two parts:

Part 1: First 5 years (8% compounded annually)

We need to calculate the future value of the investment after 5 years. Using the formula, we have:
P20,000 = PV(1 + 0.08/1)^(1*5)

Simplifying the equation, we get:
P20,000 = PV(1.08)^5

Dividing both sides by (1.08)^5, we get:
PV = P20,000 / (1.08)^5

Part 2: Next 5 years (12% compounded quarterly)

We need to calculate the future value of the investment after the first 5 years, then use it as the present value for the next 5 years. Using the formula, we have:
FV = PV(1 + 0.12/4)^(4*5)

Substituting PV with the result from Part 1, we get:
FV = (P20,000 / (1.08)^5) * (1.03)^(4*5)

Simplifying the equation, we get:
FV = (P20,000 * (1.03)^(4*5)) / (1.08)^5

Since FV (future value) is P20,000, we can substitute it and solve for PV:
P20,000 = (P20,000 * (1.03)^(4*5)) / (1.08)^5

Multiplying both sides by (1.08)^5, we get:
P20,000 * (1.08)^5 = P20,000 * (1.03)^(4*5)

Dividing both sides by P20,000, we get:
(1.08)^5 = (1.03)^(4*5)

Now, we can use a calculator or a spreadsheet to find the answer.

Using a calculator, we find that (1.08)^5 is approximately 1.4693, and (1.03)^(4*5) is approximately 1.8221.

So, we have:
1.4693 = 1.8221

This equation cannot be true, which means there is an error in the given problem or calculation. Please double-check the values provided or the calculations made.