To calculate the total gain at the end of the second year for both accounts, we can follow these steps:
- Calculate how much money is placed in the savings account and how much is placed in the stock plan.
- Calculate the final amount in the savings account after two years.
- Calculate the final amount in the stock plan after two years.
- Determine the total gain from both accounts.
Step 1: Calculate amounts in both accounts
-
Total savings: \(8500\)
-
Amount in savings account: \(40%\) of \(8500\)
\[ \text{Savings amount} = 8500 \times 0.40 = 3400 \]
-
Amount in stock plan: \(60%\) of \(8500\)
\[ \text{Stock plan amount} = 8500 \times 0.60 = 5100 \]
Step 2: Calculate final amount in savings account
The formula for the amount in a savings account with compounded interest is:
\[ A = P(1 + r)^t \]
Where:
- \(A\) is the amount of money accumulated after \(n\) years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(t\) is the number of years the money is invested or borrowed.
- For the savings account after 2 years:
- \(P = 3400\)
- \(r = 0.042\)
- \(t = 2\)
Calculating:
\[ A_{savings} = 3400(1 + 0.042)^2 \] \[ A_{savings} = 3400(1.042)^2 \] \[ A_{savings} = 3400 \times 1.086736 = 3694.50 \quad (\text{rounding to the nearest penny}) \]
Step 3: Calculate final amount in stock plan
- First year (decrease of 3%):
- Final amount after first year:
\[ A_{stock1} = 5100 \times (1 - 0.03) = 5100 \times 0.97 = 4957.00 \]
- Second year (increase of 7.5%):
- Final amount after second year:
\[ A_{stock2} = 4957.00 \times (1 + 0.075) = 4957.00 \times 1.075 = 5325.48 \quad (\text{rounding to the nearest penny}) \]
Step 4: Calculate total gain at the end of the second year
- Total amount from both accounts:
\[ \text{Total amount} = A_{savings} + A_{stock2} = 3694.50 + 5325.48 = 9019.98 \]
- Total gain:
\[ \text{Total gain} = \text{Total amount} - \text{Initial savings} = 9019.98 - 8500 = 519.98 \]
Final Result
The total gain at the end of the second year for both accounts combined is $519.98.