$7,000 is invested into two accounts: 4,000 into an account paying 5% interest compounded monthly and
$3,000 into an account paying 4.1% interest compounded continuously.
compute the total interest earned at the end of 2 years.
The formulas you require can be found here:
http://www.moneychimp.com/articles/finworks/continuous_compounding.htm
There is also a calculator to make things easier.
amount of first = 4000(1 + .05/12)^24 = 4419.76
amount in 2nd = 3000 e^(2(.05)) = 3315.51
take over
To compute the total interest earned at the end of 2 years, we will calculate the interest earned in each account separately and then add them together.
For the first account: The principal (initial investment) is $4,000 and the interest rate is 5% per year compounded monthly. The formula to calculate the compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal (initial investment)
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
In this case, P = $4,000, r = 5% = 0.05, n = 12 (compounded monthly), and t = 2 years.
Using the formula, we can calculate the future value (A) of the investment after 2 years:
A = 4000(1 + 0.05/12)^(12*2)
A = 4000(1 + 0.0041667)^(24)
A = 4000(1.0041667)^(24)
A ≈ $4,419.44
The interest earned in this account is the future value minus the principal:
Interest earned = A - P
Interest earned = 4419.44 - 4000
Interest earned ≈ $419.44
For the second account: The principal is $3,000 and the interest is 4.1% per year compounded continuously. The formula to calculate the continuous compound interest is:
A = P * e^(rt)
Where:
A = the future value of the investment
P = the principal (initial investment)
e = Euler's number (approximately 2.71828)
r = the annual interest rate (in decimal form)
t = the number of years
In this case, P = $3,000, r = 4.1% = 0.041, and t = 2 years.
Using the formula, we can calculate the future value (A) of the investment after 2 years:
A = 3000 * e^(0.041*2)
A ≈ 3000 * e^(0.082)
A ≈ 3000 * 1.0853121
A ≈ $3,255.93
The interest earned in this account is the future value minus the principal:
Interest earned = A - P
Interest earned = 3255.93 - 3000
Interest earned ≈ $255.93
Finally, we add the interest earned in both accounts:
Total interest earned = Interest earned in first account + Interest earned in second account
Total interest earned ≈ $419.44 + $255.93
Total interest earned ≈ $675.37
Therefore, the total interest earned at the end of 2 years is approximately $675.37.