Richard Simmons is selling his house. He has a choice of taking $125,000 today or $135,000 in 6 months. If he takes the money today, he can invest it at Valley Bank at 5% interest compounded monthly.
a. How much would be in the account after six months if he took the 125,000.
b. Which option should he take?
c. How much more money does he gain in making this choice?
Did this for Lynda back in December
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To calculate the answers to these questions, we need to use the compound interest formula:
A = P(1 + r/n)^(nt),
where:
A is the final amount,
P is the principal (initial amount),
r is the annual interest rate (in decimal form),
n is the number of times interest is compounded per year, and
t is the number of years.
For this problem:
P = $125,000
r = 5% = 0.05 (annual interest rate)
n = 12 (monthly compounding)
t = 6/12 (6 months converted to years)
a. To find out how much would be in the account after six months if he took $125,000, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
A = $125,000(1 + 0.05/12)^(12*(6/12))
Using a calculator or solving step-by-step:
A = $125,000(1.004166666)^6
A = $125,000(1.025)
A ≈ $128,125
After six months, if Richard took the $125,000 and invested it at Valley Bank, he would have approximately $128,125 in the account.
b. To determine which option he should take, we compare the amounts. If the amount after six months is higher than $135,000, he should take the money today; otherwise, he should wait for the $135,000.
Since $128,125 < $135,000, he should choose to take $135,000 in 6 months.
c. To calculate how much more money he gains by making this choice, we subtract the initial amount ($128,125) from the final amount ($135,000):
$135,000 - $128,125 = $6,875.
Therefore, Richard Simmons gains $6,875 by choosing to take $135,000 in 6 months instead of $125,000 today.
To answer these questions, we need to calculate the future value of the $125,000 investment after 6 months and compare it to the $135,000 he would receive after the same time.
a. To calculate the future value of the $125,000 investment after 6 months with a monthly compounding interest rate of 5%, we can use the formula for compound interest:
Future Value = Present Value * [(1 + (interest rate / n))^(n * t)]
Where:
- Present Value (PV) = $125,000
- Interest Rate (r) = 5% or 0.05 (converted to decimal)
- Number of Compounding Periods per year (n) = 12 (since it's compounded monthly)
- Time (t) = 6 months
Future Value = $125,000 * [(1 + (0.05 / 12))^(12 * 6)]
Using a calculator or spreadsheet, you can compute the Future Value to find out how much would be in the account after six months if Richard took the $125,000.
b. To determine which option Richard should take, we compare the future value calculated in part a with the $135,000 he would receive in 6 months. If the future value is higher, taking the $125,000 today and investing it is the better choice, and vice versa.
c. To calculate the difference in money gained from making either choice, subtract the initial investment ($125,000) from the higher amount (either the future value or $135,000). The result will provide the amount of additional money gained by choosing that particular option.