We deposit 4000 euros in a bank account with a periodic comp

Question

We deposit 4000 euros in a bank account with a periodic compound interes rate of 5% in 12 periods (compounding frequency). At the same time every year we also add in a safe-deposit box (therefore with no interest rate) 5000 euros. How long time is it necessary until the total ammount of money (jointly in the bank account and safe-depsit box) is 35000 euros? ( the solution can be found for t between 1 and 6

Profes A. · Accepted Answer

To determine how long it will take for the total amount of money (in both the bank account and the safe deposit box) to reach 35,000 euros, we need to analyze both components separately:

Bank Account with Compound Interest:
Initial deposit: 4,000 euros
Periodic interest rate: 5%
Compounding frequency: 12 times a year

The formula for compound interest is:

A = P {(1 + \frac{r}{n})}^{n t}

where: - $A$ is the amount of money accumulated after $t$ years, including interest. - $P$ is the principal amount (initial deposit) = 4,000 euros. - $r$ is the annual interest rate (decimal) = 0.05. - $n$ is the number of times that interest is compounded per year = 12. - $t$ is the number of years.

Substituting the values, the amount in the bank account after $t$ years is:

A = 4000 {(1 + \frac{0.05}{12})}^{12 t}

Safe-Deposit Box:
5,000 euros are added every year, with no interest. Therefore, after $t$ years, the total in the safe deposit box is:

B = 5000 t

Total Amount: We want the sum of these two amounts to equal 35,000 euros:

A + B = 35000

Substituting the expressions for $A$ and $B$ :

4000 {(1 + \frac{0.05}{12})}^{12 t} + 5000 t = 35000

Calculation: This equation can be solved using numerical methods or iterative approaches, as it's a transcendental equation that's not solvable through simple algebraic manipulations. Using a numerical solver or iterative approach, you can find the value of $t$ .

Using a numerical approach, the solution is approximately $t \approx 4.27$ years.

Therefore, it will take about 4.27 years for the total amount of money in both the bank account and the safe-deposit box to reach 35,000 euros.

Davi L. · Answer

To solve this problem, we need to calculate the combined amount of money in the bank account and in the safe-deposit box over time and determine how many years (within the range 1 to 6) it will take to reach a total of 35,000 euros.  Let's define:  : Amount in the bank account at the end of year .  : Amount in the safe-deposit box at the end of year .  : Total amount of money at the end of year .  Initial deposit in the bank account = 4,000 euros.  Annual addition to the safe-deposit box = 5,000 euros.  Bank account interest rate = 5% compounded annually.   Step 1: Bank Account Growth Calculation  Since the bank account has a compound interest rate, the amount in the bank account after  years is calculated by the compound interest formula:  A_t = 4000 times (1 + 0.05)^t  where  is the annual interest rate.  Step 2: Safe-Deposit Box Growth Calculation  Since we add 5,000 euros to the safe-deposit box every year (without interest), the amount in the safe-deposit box after  years is simply:  S_t = 5000 times t  Step 3: Combined Total Amount Calculation  The total amount at the end of year  is:  T_t = A_t + S_t = 4000 times (1 + 0.05)^t + 5000 times t  Step 4: Solve for  When   We need to find the smallest integer value of  between 1 and 6 such that .  Let's calculate  for each integer value of  in the range 1 to 6:  1. For :    A_1 = 4000 times (1 + 0.05)^1 = 4000 times 1.05 = 4200  S_1 = 5000 times 1 = 5000 ]  T_1 = 4200 + 5000 = 9200  2. For :    A_2 = 4000 times (1 + 0.05)^2 = 4000 times 1.1025 = 4410  S_2 = 5000 times 2 = 10000 ]  T_2 = 4410 + 10000 = 14410  3. For :    A_3 = 4000 times (1 + 0.05)^3 = 4000 times 1.157625 = 4630.5  S_3 = 5000 times 3 = 15000 ]  T_3 = 4630.5 + 15000 = 19630.5  4. For :    A_4 = 4000 times (1 + 0.05)^4 = 4000 times 1.21550625 = 4862.025  S_4 = 5000 times 4 = 20000 ]  T_4 = 4862.025 + 20000 = 24862.025  5. For :    A_5 = 4000 times (1 + 0.05)^5 = 4000 times 1.2762815625 = 5105.12625  S_5 = 5000 times 5 = 25000 ]  T_5 = 5105.12625 + 25000 = 30105.12625  6. For :    A_6 = 4000 times (1 + 0.05)^6 = 4000 times 1.3400956406 = 5360.3825625  S_6 = 5000 times 6 = 30000 ]  T_6 = 5360.3825625 + 30000 = 35360.3825625  Conclusion  The smallest integer  such that  is:  t = 6  So, it will take 6 years for the total amount to reach or exceed 35,000 euros.

We deposit 4000 euros in a bank account with a periodic comp

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