You invest $500 at an annual interest rate of 6.75% compounded continuously. How long will it take y our money to double?

e^(.0675t) = 2

.0675t = ln2
t = ln2/.0675 = 10.27 years

The initial amount does not affect how fast it doubles.

Well, investing money is no joke! But let's see if we can add some humor to this financial puzzle.

To find out how long it will take for your money to double, we can use the equation for continuous compound interest:

A = P * e^(rt)

Where:
A = the final amount (in this case, $1000)
P = the initial principal (in this case, $500)
e = a mathematical constant approximately equal to 2.71828
r = the annual interest rate (in this case, 6.75% or 0.0675 in decimal form)
t = the time (what we're trying to find)

So let's put it all together and solve for t:

1000 = 500 * e^(0.0675t)

Now, let me throw in a quick joke to keep the conversation light: Why did the investor bring a ladder to the bank? Because they wanted to reach that high interest rate!

Back to the equation! We need to isolate t, so let's divide both sides of the equation by 500:

2 = e^(0.0675t)

Now, we can take the natural logarithm of both sides to get rid of the exponential:

ln(2) = 0.0675t

Finally, we can solve for t by dividing both sides by 0.0675:

t = ln(2)/0.0675

And there you have it! Using a calculator, you'll find that t is approximately 10.264 years. So it will take around 10.264 years for your money to double at an annual interest rate of 6.75% compounded continuously.

To find out how long it takes for your money to double, you can use the formula for continuous compound interest:

A = P * e^(rt)

Where:
A = final amount (2 times the initial amount, since you want to double the money)
P = principal amount (initial investment of $500)
r = annual interest rate (6.75% or 0.0675 as a decimal)
t = time in years

Substituting the values into the formula, we get:

2P = P * e^(0.0675 * t)

Dividing both sides of the equation by P, we have:

2 = e^(0.0675 * t)

To solve for t, we take the natural logarithm of both sides of the equation:

ln(2) = 0.0675 * t

Dividing both sides of the equation by 0.0675, we get:

t = ln(2) / 0.0675

Using a calculator, the value of ln(2) is approximately 0.6931. Substituting this value into the equation, we have:

t = 0.6931 / 0.0675

t ≈ 10.2698

Therefore, it will take approximately 10.27 years (rounded to two decimal places) for your money to double with an annual interest rate of 6.75% compounded continuously.

To find out how long it will take for your money to double, we can use the formula for continuous compound interest:

A = P * e^(rt),

where:
A is the final amount,
P is the principal investment amount,
e is the base of the natural logarithm (approximately 2.71828),
r is the annual interest rate, and
t is the time in years.

In this case, our initial investment amount, P, is $500, and we want to find out how long it will take for it to double, so our final amount, A, would be $1000.

Plugging in these values into the formula, we have:

1000 = 500 * e^(0.0675t).

Now, we need to solve this equation for t.

First, divide both sides of the equation by 500:

2 = e^(0.0675t).

Next, take the natural logarithm (ln) of both sides:

ln(2) = ln(e^(0.0675t)).

Since ln(e^x) = x, we can simplify the equation further:

ln(2) = 0.0675t.

Now, divide both sides of the equation by 0.0675:

t = ln(2) / 0.0675.

Calculating this, we find:

t ≈ 10.28 years.

Therefore, it will take approximately 10.28 years for your money to double at an annual interest rate of 6.75% compounded continuously.