on the day Rachel was born, her grandparents deposited $ 500 into a savings account that earns 4.8%/a compounded monthly . They deposited the same amount on her 5th, 15th, and 10th birthdays. determine the balance in the account on Rachel's 18th birthday
amount = 500[ ( 1.004)^216+1.004^156 + 1.004^96 + 1.004^36 ]
= .....
To determine the balance in the account on Rachel's 18th birthday, we need to calculate the future value of each deposit separately and then sum them up.
First, let's calculate the future value of the initial deposit of $500. We'll use the compound interest formula:
FV = PV * (1 + r/n)^(n*t)
Where:
FV = Future Value
PV = Present Value (initial deposit)
r = Annual interest rate (converted to decimal)
n = Number of times interest is compounded per year
t = Number of years
In this case, PV = $500, r = 4.8% (or 0.048 as a decimal), n = 12 (compounded monthly), and t = 18 (years):
FV1 = $500 * (1 + 0.048/12)^(12*18)
Calculating FV1:
FV1 = $500 * (1 + 0.004)^216
FV1 ≈ $500 * (1.004)^216
FV1 ≈ $500 * 3.820465
FV1 ≈ $1,910.23 (rounded to the nearest cent)
Next, let's calculate the future value of the deposits made on Rachel's 5th, 10th, and 15th birthdays. Since they were all equal to the initial deposit of $500, we can use the same formula:
FV2 = PV * (1 + r/n)^(n*t)
In this case, PV = $500, r = 4.8% (or 0.048 as a decimal), n = 12 (compounded monthly), and t = 13 (from 5th birthday to 18th birthday):
FV2 = $500 * (1 + 0.004)^156
FV2 ≈ $500 * (1.004)^144
FV2 ≈ $500 * 2.597788
FV2 ≈ $1,298.89 (rounded to the nearest cent)
Since the deposits made on the 5th, 10th, and 15th birthdays are the same, the total future value of these three deposits is:
3 * FV2 ≈ 3 * $1,298.89
Total Future Value of the three deposits ≈ $3,896.67 (rounded to the nearest cent)
Now, let's calculate the overall balance by summing up the future value of the initial deposit and the future value of the three additional deposits:
Overall Balance = FV1 + Total Future Value of the three deposits
Overall Balance ≈ $1,910.23 + $3,896.67
Overall Balance ≈ $5,806.90
Therefore, the balance in the account on Rachel's 18th birthday is approximately $5,806.90.
To determine the balance in Rachel's savings account on her 18th birthday, we will need to calculate the future value of each deposit and add them together.
First, let's calculate the future value of the initial deposit of $500. The formula to calculate future value with compound interest is:
FV = P(1 + r/n)^(nt)
Where:
FV = future value
P = principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case, the principal amount (P) is $500, the annual interest rate (r) is 4.8% (or 0.048 as a decimal), the interest is compounded monthly, so n is 12, and t is the number of years (18 - the year Rachel was born, assuming her 18th birthday is her 18th full year).
Using these values, we can calculate the future value of the initial deposit:
FV1 = $500(1 + 0.048/12)^(12*(18 - birth year))
Next, let's calculate the future values of the deposits made on her 5th, 10th, and 15th birthdays. Since each deposit is the same, the calculations are identical. We'll refer to these future values as FV2, FV3, and FV4, respectively.
FV2 = $500(1 + 0.048/12)^(12*(18 - 5))
FV3 = $500(1 + 0.048/12)^(12*(18 - 10))
FV4 = $500(1 + 0.048/12)^(12*(18 - 15))
Finally, we can determine the balance in the account on Rachel's 18th birthday by summing up all the future values:
Balance = FV1 + FV2 + FV3 + FV4
Substituting the calculated values, you can determine the balance in the account on Rachel's 18th birthday.