Reiny
answered
i = .054/4 = .0135
n = ?
Amount = 250000
5000(1.0135^n - 1)/.01535 = 250000
1.0135^n - 1 = .0135(50000) = .675
1.0135^n = 1.675
take log of both sides and use log rules
n log 1.0135 = log 1.675
n = 38.47 quarter years or appr 9 years and 7 months
Step-by-Step Bot
answered
FV = P * (1 + r/n)^(nt)
Where:
FV = Future Value (amount Alonzo wants to accumulate, $250,000)
P = Payment made every quarter ($5,000)
r = Interest rate (5.4%, or 0.054)
n = Number of times the interest is compounded per year (4 times, as there are 4 quarters in a year)
t = Number of years
Plugging in the values we have:
250000 = 5000 * (1 + 0.054/4)^(4t)
Simplifying the equation:
(1 + 0.054/4)^(4t) = 250000/5000
(1 + 0.0135)^(4t) = 50
Now, we can solve for t by taking the logarithm of both sides of the equation:
4t * log(1.0135) = log(50)
t = log(50) / (4 * log(1.0135))
Using a calculator, we find:
t ≈ 19.875
Therefore, it will take Alonzo approximately 19.875 years to retire. Rounded to the nearest year, this is 20 years.
Explain Bot
answered
A = P * ((1 + r/n)^(nt) - 1) / (r/n)
Where:
A = accumulated value (target amount)
P = payment per period
r = interest rate
n = number of compounding periods per year
t = number of years
In this case, Alonzo plans to accumulate $250,000, with quarterly payments of $5,000. The interest rate is 5.4%, compounded quarterly.
Let's plug in the values into the formula:
250,000 = 5,000 * ((1 + 0.054/4)^(4t) - 1) / (0.054/4)
Simplifying further:
(1 + 0.054/4)^(4t) - 1 = 250,000 * (0.054/4) / 5,000
Now, we can solve for t using logarithms.
(1 + 0.054/4)^(4t) = (250,000 * (0.054/4) / 5,000) + 1
Taking the natural logarithm on both sides:
ln((1 + 0.054/4)^(4t)) = ln((250,000 * (0.054/4) / 5,000) + 1)
Now, we can bring the exponent down:
(4t) * ln(1 + 0.054/4) = ln((250,000 * (0.054/4) / 5,000) + 1)
Finally, solving for t:
t = ln((250,000 * (0.054/4) / 5,000) + 1) / (4 * ln(1 + 0.054/4))
Using a calculator, we can find the value of t to the nearest year.
