Determine the amount of money that will be accumulated in an account that pays compound interest, given the initial principal of $ 29,400 invested at 2.77% annual interest for 7 years compounded
(a) daily (n365);
(b) continuously.
To calculate the amount of money accumulated, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated
P = the initial principal (or amount invested)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
(a) For daily compounding (n = 365), the formula becomes:
A = 29,400(1 + 0.0277/365)^(365*7)
Calculating this formula gives:
A ≈ $33977.37
Therefore, the amount of money accumulated in the account after 7 years with daily compounding is approximately $33,977.37.
(b) For continuous compounding, the formula becomes:
A = P * e^(rt)
Where e is Euler's number, approximately 2.71828.
A = 29,400 * e^(0.0277*7)
Calculating this formula gives:
A ≈ $33,930.04
Therefore, the amount of money accumulated in the account after 7 years with continuous compounding is approximately $33,930.04.
To determine the amount of money accumulated in an account that pays compound interest, we can use the compound interest formula:
A = P(1 + r/n)^(n*t)
Where:
A = Accumulated amount of money
P = Initial principal (amount invested)
r = Annual interest rate (expressed as a decimal)
n = Number of times interest is compounded per year
t = Number of years
(a) For daily compounding (n = 365), let's calculate the accumulated amount:
A = 29,400(1 + 0.0277/365)^(365*7)
First, divide the annual interest rate by the number of compounding periods in a year:
0.0277/365 ≈ 0.000076
Next, calculate the exponent:
365*7 = 2,555
Now, substitute the values into the formula and solve for A:
A = 29,400(1 + 0.000076)^(2,555)
Using a calculator, raise the expression in parentheses to the power of 2,555, then multiply it by the initial principal:
A ≈ 29,400(1.000076)^2555 ≈ $33,885.76
Therefore, the amount of money accumulated in the account with daily compounding is approximately $33,885.76.
(b) For continuous compounding, we can use the formula:
A = P * e^(r*t)
Where:
e = Euler's number (approximately 2.71828)
Let's calculate the accumulated amount considering continuous compounding:
A = 29,400 * e^(0.0277*7)
Multiply the annual interest rate by the number of years:
0.0277*7 ≈ 0.1939
Next, raise Euler's number (e) to the power of the result:
e^0.1939 ≈ 1.2133
Now, multiply the result by the initial principal:
A ≈ 29,400 * 1.2133 ≈ $35,684.62
Therefore, the amount of money accumulated in the account with continuous compounding is approximately $35,684.62.
To determine the amount of money accumulated in an account with compound interest, we can use the formula:
A = P(1 + r/n)^(n*t)
Where:
A = the amount accumulated
P = the principal (initial investment)
r = the annual interest rate (expressed as a decimal)
n = the number of times interest is compounded per year
t = the number of years
(a) Daily compounding (n = 365):
Using the given information:
P = $29,400
r = 2.77% = 0.0277 (expressed as a decimal)
n = 365
t = 7
Substituting these values into the formula:
A = $29,400(1 + 0.0277/365)^(365*7)
Calculating the expression inside parentheses:
(1 + 0.0277/365) = 1.0000758039
Now, calculating the final value of A:
A = $29,400 * (1.0000758039)^(365*7)
A ≈ $29,400 * 1.2070799788
A ≈ $35,485.48
Therefore, the amount of money accumulated in the account with daily compounding after 7 years is approximately $35,485.48.
(b) Continuous compounding:
Using the given information:
P = $29,400
r = 2.77% = 0.0277 (expressed as a decimal)
t = 7
To calculate for continuous compounding, we use the formula:
A = P * e^(r*t)
Where e is the base of the natural logarithm (approximately 2.71828).
Substituting the values into the formula:
A = $29,400 * e^(0.0277*7)
Calculating the exponential term:
e^(0.0277*7) ≈ 1.1915216656
Calculating the final value of A:
A = $29,400 * 1.1915216656
A ≈ $35,020.59
Therefore, the amount of money accumulated in the account with continuous compounding after 7 years is approximately $35,020.59.