3) Your uncle has $280,000 invested at 7.5%, and he now wants to retire. He wants to withdraw $35,000 at the end of each year, starting at the end of this year. He also wants to have $25,000 left to give you when he ceases to withdraw funds from the account. For how many years can he make the $35,000 withdrawals and still have $25,000 left in the end?
To find out how many years your uncle can make $35,000 withdrawals and still have $25,000 left in the end, we need to set up a calculation based on his investment and withdrawal amounts.
First, let's break down the problem into steps:
1. Calculate the annual interest earned on the $280,000 investment at a rate of 7.5%.
2. Subtract the $35,000 withdrawal made at the end of the year.
3. Add the calculated interest earned on the remaining balance.
4. Repeat steps 2 and 3 until the remaining balance reaches $25,000 or less.
Now let's go through these steps in detail:
Step 1: Calculate the annual interest earned on the $280,000 investment at a rate of 7.5%.
To calculate the annual interest, multiply the investment amount by the interest rate:
$280,000 * 7.5% = $21,000
Step 2: Subtract the $35,000 withdrawal made at the end of the year.
$280,000 - $35,000 = $245,000
Step 3: Add the calculated interest earned on the remaining balance.
$245,000 + $21,000 = $266,000
Step 4: Repeat steps 2 and 3 until the remaining balance reaches $25,000 or less.
Continue with the same process for the following years until the remaining balance is no longer above $25,000. Keep subtracting $35,000 and adding the annual interest earned.
Let's calculate the remaining balance for each year until it reaches $25,000 or less:
Year 1: $266,000 - $35,000 + $19,950 (7.5% of $266,000) = $250,950
Year 2: $250,950 - $35,000 + $18,821.25 (7.5% of $250,950) = $234,771.25
Year 3: $234,771.25 - $35,000 + $17,608.34 (7.5% of $234,771.25) = $217,379.59
Year 4: $217,379.59 - $35,000 + $16,303.47 (7.5% of $217,379.59) = $198,683.06
Year 5: $198,683.06 - $35,000 + $14,900.73 (7.5% of $198,683.06) = $178,583.79
After the 5th year, the remaining balance is $178,583.79, which is less than $25,000. Therefore, your uncle can make $35,000 withdrawals for 5 years and still have $25,000 left in the end.
To determine the number of years your uncle can make $35,000 withdrawals and still have $25,000 left at the end, we can use a formula for the present value of an annuity.
The present value of an annuity formula is:
PV = P * [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present value of the annuity
P = Payment amount per period
r = Interest rate per period
n = Number of periods
In this case, we want to find the value of n.
Let's plug in the values we have:
PV = $25,000
P = $35,000
r = 7.5% or 0.075 per year
Substituting these values in, the formula becomes:
$25,000 = $35,000 * [(1 - (1 + 0.075)^(-n)) / 0.075]
Now, let's solve for n.
First, divide both sides of the equation by $35,000:
$25,000 / $35,000 = [(1 - (1 + 0.075)^(-n)) / 0.075]
Next, multiply both sides of the equation by 0.075:
(0.7143) = [(1 - (1 + 0.075)^(-n)) / 0.075]
Now, subtract 1 from both sides of the equation:
-0.2857 = -(1 + 0.075)^(-n)
Since we have a negative sign on both sides, we can eliminate them:
0.2857 = (1 + 0.075)^(-n)
Take the reciprocal of both sides to get rid of the exponent:
1 / 0.2857 = (1 + 0.075)^n
Now, let's calculate the right-hand side:
3.5 = (1.075)^n
To solve for n, we can take the logarithm of both sides:
log(3.5) = log(1.075)^n
Using logarithm properties, we can bring the exponent down:
log(3.5) = n * log(1.075)
Now, divide both sides by log(1.075):
n = log(3.5) / log(1.075)
Using a calculator, we find that n is approximately equal to 9.9 years.
Therefore, your uncle can make $35,000 withdrawals for approximately 9.9 years and still have $25,000 left in the end.