A person deposits 7000 in a bank account. Three years later, he withdraws 5000.
Two years after that, he withdraws an additional 3000. One year after that, he deposits an additional 4000. If interest is 6% compounded annually and that no other deposits or withdrawals are made, how much in his account, 10 years after initial deposits?
(a) 9586 (b) 6053 (c) 3000 (d) none
To solve this problem, let's break it down into several steps.
Step 1: Calculate the value after 3 years
Using the compound interest formula, we can find the value after 3 years:
A = P(1+r)^n
Where:
A = value after n years
P = principal amount (initial deposit)
r = interest rate (6% or 0.06)
n = number of years (3)
A = 7000(1+0.06)^3
A = 7000(1.06)^3
A = 7000(1.191016)
A = 8347.11
Step 2: Calculate the value after 5 years
To find the value after 5 years, we need to subtract the first withdrawal of $5000:
Value after 5 years = 8347.11 - 5000
Value after 5 years = 3347.11
Step 3: Calculate the value after 7 years
To find the value after 7 years, we need to subtract the second withdrawal of $3000:
Value after 7 years = 3347.11 - 3000
Value after 7 years = 347.11
Step 4: Calculate the value after 8 years
To find the value after 8 years, we need to add the additional deposit of $4000:
Value after 8 years = 347.11 + 4000
Value after 8 years = 4347.11
Step 5: Calculate the value after 10 years
Using the compound interest formula again, we can find the value after 10 years:
A = 4347.11(1+0.06)^2
A = 4347.11(1.06)^2
A = 4347.11(1.1236)
A = 4887.24
Therefore, the person has $4887.24 in their account 10 years after the initial deposit.
So, the correct answer is (d) none.
To solve this problem, we can break it down into three steps:
Step 1: Calculate the amount after 10 years with no withdrawals.
Step 2: Calculate the amount after the first withdrawal.
Step 3: Calculate the final amount after the second withdrawal and additional deposit.
Step 1:
We can use the compound interest formula: A = P(1 + r/n)^(nt), where A is the future amount, P is the principal (initial deposit), r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.
In this case, the initial deposit is $7,000.
A1 = 7000(1 + 0.06/1)^(1*10)
A1 = 7000(1 + 0.06)^10
A1 ≈ 7000 * 1.790845
A1 ≈ 12,536.92
So, after 10 years with no withdrawals, the amount in the account is approximately $12,536.92.
Step 2:
Now, we need to calculate the amount after the first withdrawal of $5,000 from the initial amount calculated in step 1.
A2 = A1 - 5000
A2 ≈ 12,536.92 - 5000
A2 ≈ 7,536.92
So, after the first withdrawal, the amount in the account is approximately $7,536.92.
Step 3:
Next, we need to calculate the amount after the second withdrawal of $3,000 and the additional deposit of $4,000.
First withdrawal:
A3 = A2 - 3000
A3 ≈ 7,536.92 - 3000
A3 ≈ 4,536.92
Additional deposit:
A4 = A3 + 4000
A4 ≈ 4,536.92 + 4000
A4 ≈ 8,536.92
So, after the second withdrawal and additional deposit, the amount in the account is approximately $8,536.92.
Therefore, the correct answer is option (d) none, as none of the provided answer choices matches the final amount of $8,536.92.
To solve this question, we need to calculate the amount in the bank account after 10 years. Let's break down the steps to get the answer:
Step 1: Calculate the amount after 3 years.
Since the interest is compounded annually, we can use the compound interest formula: A = P(1 + r/n)^(nt), where A is the amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the time in years.
In this case, the principal is $7000, the interest rate is 6% (or 0.06 in decimal form), the time is 3 years, and the interest is compounded annually (n = 1).
Plugging in these values, we get: A = 7000(1 + 0.06/1)^(1*3) = 7000(1.06)^3 = 7000 * 1.191016 = 8347.11.
Step 2: Subtract the first withdrawal after 3 years.
Subtract $5000 from the amount calculated above: 8347.11 - 5000 = 3347.11.
Step 3: Calculate the amount after 5 years.
We now need to calculate the amount after an additional 2 years. Using the formula mentioned in Step 1, we have: A = 3347.11(1 + 0.06/1)^(1*2) = 3347.11(1.06)^2 = 3347.11 * 1.1236 = 3756.15.
Step 4: Subtract the second withdrawal after 5 years.
Subtract $3000 from the amount calculated above: 3756.15 - 3000 = 756.15.
Step 5: Calculate the amount after 6 years.
We now need to calculate the amount after an additional 1 year. Using the formula mentioned in Step 1, we have: A = 756.15(1 + 0.06/1)^(1*1) = 756.15(1.06)^1 = 756.15 * 1.06 = 801.13.
Step 6: Add the third deposit after 6 years.
Add $4000 to the amount calculated above: 801.13 + 4000 = 4801.13.
Step 7: Calculate the amount after 10 years.
We now need to calculate the amount after an additional 4 years. Using the formula mentioned in Step 1, we have: A = 4801.13(1 + 0.06/1)^(1*4) = 4801.13(1.06)^4 = 4801.13 * 1.26248 = 6053.20.
Therefore, the amount in the bank account, 10 years after the initial deposit, is approximately $6053.20.