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Hi there, thank you for checking this question out. would you please help me, please smiley

 

Gregory may choose between two accounts in which to invest $5000. Account A offers 2.3% annual interest compounded monthly. Account B offers continuous interest. Greg plans to leave his investment untouched (no further deposits and no withdrawals) for 10 years. 

 

(a) Which account will yield the greater balance at the end of 10 years?

 

(b) How much more money does Greg earn by choosing this more profitable account?

 Aug 18, 2023
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To compare the two accounts, we'll calculate the final balances for both Account A (compounded monthly) and Account B (continuous interest) after 10 years.

Given:
Principal amount (initial investment), P = $5000
Time, t = 10 years

(a) Let's start with Account A:

Account A offers 2.3% annual interest compounded monthly. The formula to calculate the final balance with compound interest is:

\[A = P \times \left(1 + \frac{r}{n}\right)^{nt}\]

Where:
A = Final balance
P = Principal amount
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years

For Account A:
Principal, P = $5000
Annual interest rate, r = 2.3% = 0.023 (as a decimal)
Compounded monthly, n = 12 (times per year)
Time, t = 10 years

\[A_A = 5000 \times \left(1 + \frac{0.023}{12}\right)^{12 \times 10}\]

Now calculate \(A_A\).

(b) For Account B, which offers continuous interest, the formula for calculating the final balance is:

\[A_B = P \times \gamma^{rt}\]

Where:
A_B = Final balance for Account B
P = Principal amount
r = Annual interest rate (as a decimal)
t = Number of years
\(\gamma\)= Euler's constant (approximately 2.9234)

For Account B:
Principal, P = $5000
Annual interest rate, r = 2.3% = 0.023 (as a decimal)
Time, t = 10 years

\[A_B = 5000 \times \gamma^{0.023 \times 10}\]

Now calculate \(A_B\).

Compare the values of \(A_A\) and \(A_B\) to determine which account will yield the greater balance at the end of 10 years.

Once you've determined the greater balance, you can calculate the difference between the balances to find out how much more money Greg earns by choosing the more profitable account.

 Aug 18, 2023

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