Ahmed's Savings Account Growth and Fund Transfer Analysis

Ahmed deposits $4,000 in a savings account that pays 8% interest compounded monthly. Three years later, he deposits $5,000. Two years after the $5,000 deposit he makes another deposit in the amount of $7,000. Four years after the $7,000 deposit, half of the accumulated money is transferred to a fund that pays 8% interest compounded quarterly. How much money will be in each account six years after the transfer?

To solve this problem, we need to calculate the amount of money in Ahmed's savings account at each stage and then determine the final amounts in both accounts after the transfer. We'll use the formula for compound interest:

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

where:

• ( A ) is the amount of money accumulated after ( t ) years, including interest.
• ( P ) is the principal amount (the initial amount of money).
• ( r ) is the annual interest rate (decimal).
• ( n ) is the number of times that interest is compounded per year.
• ( t ) is the time the money is invested for in years.

Step-by-Step Calculation:

1. Initial Deposit of $4,000:

   • Principal (( P )) = $4,000
   • Annual interest rate (( r )) = 8% = 0.08
   • Compounded monthly (( n )) = 12
   • Time (( t )) = 3 years

   [ A_1 = 4000 \left(1 + \frac{0.08}{12}\right)^{12 \times 3} ]

   [ A_1 = 4000 \left(1 + 0.0066667\right)^{36} ]

   [ A_1 = 4000 \left(1.0066667\right)^{36} ]

   [ A_1 \approx 4000 \times 1.26824 \approx 5072.96 ]

2. Deposit of $5,000 after 3 years:

   • Principal (( P )) = $5,000
   • Time (( t )) = 2 years

   [ A_2 = 5000 \left(1 + \frac{0.08}{12}\right)^{12 \times 2} ]

   [ A_2 = 5000 \left(1 + 0.0066667\right)^{24} ]

   [ A_2 = 5000 \left(1.0066667\right)^{24} ]

   [ A_2 \approx 5000 \times 1.16986 \approx 5849.30 ]

3. Deposit of $7,000 after 5 years:

   • Principal (( P )) = $7,000
   • Time (( t )) = 4 years

   [ A_3 = 7000 \left(1 + \frac{0.08}{12}\right)^{12 \times 4} ]

   [ A_3 = 7000 \left(1 + 0.0066667\right)^{48} ]

   [ A_3 = 7000 \left(1.0066667\right)^{48} ]

   [ A_3 \approx 7000 \times 1.43236 \approx 10026.52 ]

4. Total amount before transfer: [ A_{\text{total}} = A_1 + A_2 + A_3 ]

   [ A_{\text{total}} \approx 5072.96 + 5849.30 + 10026.52 \approx 20948.78 ]

5. Transfer half to a new fund:

   • Amount transferred = ( \frac{20948.78}{2} \approx 10474.39 )
   • Remaining in the original account = ( 10474.39 )

6. Calculate the amount in each account after 6 years:

   Original Account (compounded monthly):

   • Principal (( P )) = $10,474.39
   • Time (( t )) = 6 years

   [ A_{\text{original}} = 10474.39 \left(1 + \frac{0.08}{12}\right)^{12 \times 6} ]

   [ A_{\text{original}} = 10474.39 \left(1 + 0.0066667\right)^{72} ]

   [ A_{\text{original}} = 10474.39 \left(1.0066667\right)^{72} ]

   [ A_{\text{original}} \approx 10474.39 \times 1.60103 \approx 16768.12 ]

   New Fund (compounded quarterly):

   • Principal (( P )) = $10,474.39
   • Compounded quarterly (( n )) = 4

   [ A_{\text{new}} = 10474.39 \left(1 + \frac{0.08}{4}\right)^{4 \times 6} ]

   [ A_{\text{new}} = 10474.39 \left(1 + 0.02\right)^{24} ]

   [ A_{\text{new}} = 10474.39 \left(1.02\right)^{24} ]

   [ A_{\text{new}} \approx 10474.39 \times 1.60103 \approx 16768.12 ]

Final Amounts:

• Original Account: $16,768.12
• New Fund: $16,768.12

So, six years after the transfer, there will be approximately $16,768.12 in each account.