SOLUTION: You deposit $4000 into an account that earns 5% compounded annually. A friend deposits $3750 into an account that earns 4.95% annual interest, compounded continuously. Will your fr

Algebra ->  Customizable Word Problem Solvers  -> Finance -> SOLUTION: You deposit $4000 into an account that earns 5% compounded annually. A friend deposits $3750 into an account that earns 4.95% annual interest, compounded continuously. Will your fr      Log On

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Question 1140028: You deposit $4000 into an account that earns 5% compounded annually. A friend deposits $3750 into an account that earns 4.95% annual interest, compounded continuously. Will your friend's balance ever equal yours? If so, when?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
f = p * (1 + r) ^ n

f is the future value.
p is the present value.
r is the interest rate per time period.
n is the number of time periods.

that's the formula for discrete compounding.

the formula for continuous compounding is f = p * e ^ (r * n)

f is the future value.
p is the present value.
e is the scientific constant equal to 2.718281828.....
it is shown as the e ^ x key on your calculator.
r is the interest rate per time period.
n is the number of time periods.

their balance will be the same when the future value of their respective accounts are equal to each other.

this occurs when p * (1 + r) ^ n is equal to p * e ^ (r * n)

given your inputs, the formulas become:

4000 * (1 + .05) ^ n = 3750 * e ^ (.0495 * n)

divide both sides of this equation by 3750 to get:

4000 / 3750 * (1 + .05) ^ n = e ^ (.0495 * n)

divide both sides of this equation by (1 + .05) ^ n to get:

4000 / 3750 = (e ^ (.0495 * n) / ((1 + .05) ^ n)

take the natural log of both sides of this equation to get:

ln(4000 / 3750) = ln((e ^ (.0495 * n) / ((1 + .05) ^ n))

since ln(a/b) is equal to ln(a) - ln(b), your equation becomes:

ln(4000 / 3750) = ln(e ^ (.0495 * n)) - ln((1 + .05) ^ n)

since ln(a^b) = b * ln(a), your equation becomes:

ln(4000 / 3750) = .0495 * n * ln(e) - n * ln(1 + .05)

since ln(e) is equal to 1, your equation becomes:

ln(4000 / 3750) = .0495 * n - n * ln(1 + .05)

factor out the n to get:

ln(4000 / 3750) = n * (.0495 - ln(1 + .05))

solve for n to get:

n = ln(4000 / 3750) / (.0495 - ln(1 + 05) = 90.92034856.

the balance in both accounts will be equal in 90.92034856 years.

4000 * (1 + .05) ^ 90.92034856 = 337,752.4038

3750 * e ^ (.0495 * 90.92034856) = 3347,752.4038

your solution is that the account balances will be equal in 90.92034856 years.

the steps involved might be easier to see in my hand drawn worksheet shown below.

$$$

the transition from step 3 to step 4 takes advantages of the fact that ln(a/b) = ln(a) - ln(b).

the transition from step 4 to step 5 takes advantage of the fact that ln(a^b) = b * ln(a) and also takes advantage of the fact that ln(e) = 1.

that allows ln(e^(.0495*n)) to become equal to .0495 * n * ln(e) which then becomes equal to .0495 * n.

that also allows ln(1.05 ^ n) to become equal to n * ln(1.05).

step 7 factors out the n and then divides both sides of the equation by (.0495 - ln(1.05).

step 8 shows the result.

note that (1 + .05) is the same as 1.05.

showing 1.05 as (1 + .05) is done to reinforce the concept that it comes from the general expression of (1 + r).