Question

You invest $2500 in an account to save for college. account 1 pays 6% annual interest compounded quarterly. account 2 pays 4% annual interest compounded continuously. after 10 years, what will be the difference in the value of the two accounts?

A) Account 1 will have a higher value.
B) Account 2 will have a higher value.
C) Both accounts will have the same value.
D) More information is needed to determine the difference.

Answer 1

After 10 years, calculate
`\(A_1\)` and
`\(A_2\)` using compound interest formulas. If
`\(A_1 > A_2\),` the answer is A; if
`\(A_2 > A_1\),` the answer is B; if
`\(A_1 = A_2\),` the answer is C.

Let's calculate the future value of the investment in both accounts after 10 years and compare the results:

Account 1 (6% compounded quarterly):

The formula for compound interest is
`\(A = P\left(1 + (r)/(n)\right)^(nt)\),` where:

-A is the future value of the investment,

-P is the principal amount (initial investment),

-r is the annual interest rate (as a decimal),

-n is the number of times interest is compounded per year, and

-t is the time the money is invested for in years.

For Account 1:

P = 2500,

r = 0.06,

n = 4 compounded quarterly,

t = 10 years.

`\[A_1 = 2500\left(1 + (0.06)/(4)\right)^(4 * 10)\]`

Account 2 (4% compounded continuously):

The formula for continuously compounded interest is
`\(A = Pe^(rt)\),`where:

- e is the mathematical constant approximately equal to 2.71828.

For Account 2:

P = 2500,

r = 0.04,

t = 10 years.

`\[A_2 = 2500 * e^(0.04 * 10)\]`

After calculating
`\(A_1\)` and
`\(A_2\),` compare the values. If
`\(A_1 > A_2\),` then the correct answer is A. If
`\(A_2 > A_1\)`, then the correct answer is B. If
`\(A_1 = A_2\),` then the correct answer is C.