If $800 is deposited into an account that pays 6%/a compounded monthly, how many months will it take the investment to grow to $2000?

Question

Tastefully illustrate a financial concept in an image. As per the concept, there is an old, ornate piggy bank next to a luxurious mahogany desk. On the desk, there is an account book displaying neat columns of numbers and an old-fashioned desk calculator. The backdrop shows a calendar with changing pages to signify passing months. Around the scene, there is a subtle sense of growth, represented by a small tree sprouting from a ceramic pot, and progressively getting taller and fuller.

If $800 is deposited into an account that pays 6%/a compounded monthly, how many months will it take the investment to grow to $2000?

a)48
b)184
c)188
d)333

Answer 1

You need

800(1 + 0.06/12)^n = 2000
now solve for n. It will take a little over 15 years.

Answer 2

To solve this problem, we can use the formula for compound interest:

A = P(1 + r/n)^(n*t)

Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years

In this case, the principal amount (P) is $800, the annual interest rate (r) is 6% or 0.06, and the interest is compounded monthly, so the number of times compounded per year (n) is 12. We want to find the number of months (t) it will take for the investment to grow to $2000.

Let's plug in the given values into the formula:

2000 = 800(1 + 0.06/12)^(12*t)

Now, we can simplify the equation:

2.5 = (1 + 0.005)^12t

Taking the natural logarithm (ln) of both sides:

ln(2.5) = ln[(1 + 0.005)^12t]

Using the property of logarithms, we can bring the exponent down:

ln(2.5) = 12t * ln(1 + 0.005)

Dividing both sides by 12 * ln(1 + 0.005):

t = ln(2.5) / (12 * ln(1.005))

Using a calculator, we can compute the value of t:

t ≈ 188.24

Rounding to the nearest whole number, it will take approximately 188 months for the investment to grow to $2000.

Therefore, the correct answer is option c) 188.

Answer 3

To find the number of months it will take for the investment to grow to $2000, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where:
A = final amount (in this case, $2000)
P = principal amount (initial deposit, in this case, $800)
r = annual interest rate (6% converted to decimal form, 0.06)
n = number of times interest is compounded per year (monthly, so 12)
t = time in years

We need to solve for t, the number of years.

Let's substitute the known values into the formula:
$2000 = $800(1 + 0.06/12)^(12t)

Now, divide both sides of the equation by $800:
2.5 = (1 + 0.06/12)^(12t)

Next, let's take the logarithm of both sides to solve for t:

log(2.5) = log((1 + 0.06/12)^(12t))

Using the logarithm property, we can bring the exponent down:

log(2.5) = 12t * log(1 + 0.06/12)

Now, divide both sides by 12 times log(1 + 0.06/12):

t = log(2.5)/(12 * log(1 + 0.06/12))

Using a calculator, we can evaluate the right side to get the value of t.

t ≈ 48.006

Therefore, it will take approximately 48 months (4 years) for the investment to grow to $2000.

So, the correct answer is option (a) 48.

Answer 4

2000 = 800 [1 + (.06 / 12)]^m

log(2000 / 800) = m log(1.005)

m = log(2.5) / log(1.005)

Answer 5

Oh, calculations... my arch-nemesis! Let's see if we can spruce things up a bit with some humor, shall we?

Well, the investment starts at $800, and we want it to grow to $2000. That's quite the growth spurt, huh? Like a beanstalk reaching for the sky!

Now, with a 6% interest rate, compounded monthly, we'll have to do some calculations. But don't worry, I'll try to make this as amusing as possible!

To find out how many months it will take, we need to use a magical formula called logarithm - I'm more of a joker than a mathematician, so bear with me!

Using the compound interest formula:

A = P(1 + r/n)^(nt)

Where:
A is the amount we want to reach ($2000),
P is the initial principal ($800),
r is the annual interest rate (6% or 0.06),
n is the number of times the interest gets compounded per year (12 times for monthly),
t is the time in years (which we're trying to find).

Now, if we plug in the numbers, let's see what we get. *drumroll, please*

2000 = 800(1 + 0.06/12)^(12t)

Voilà! Now we just need to solve for t. Applying the power of logarithms... *abracadabra*

t = log(2000/800) / (12 * log(1 + 0.06/12))

And after some number crunching, we get t ≈ 47.9969. But hey, we can't have a partial month, can we? So we need to round up!

Therefore, the investment will take approximately 48 months to grow to $2000. Ding, ding, ding! Answer choice A takes the prize!

Now, if only money could grow on trees like those magic beans, we'd all be laughing our way to millionaire status!