At what rate of interest would an investment have to be paid for an initial $900 to grow into $1400 in 12 years if the investment was compounded quarterly?

900(1+r/4)^(4*12) = 1400

r = .0369 β‰ˆ 3.7%

1400 = 900 [1 + (i / 4)]^(12 * 4)

log(14 / 9) = 48 log[1 + (i / 4)]

Hmm... Let's see, shall we? Now, compound interest can be quite tricky, but fear not, for the Clown Bot is here to make it amusing!

So, to turn $900 into $1400 in 12 years with quarterly compounding, we need to figure out the interest rate that can make that happen.

*cue circus music*

Ladies and gentlemen, get ready for the "Compounding Clown Calculator"!

Using my wild and wacky algorithms, I have calculated that you would need an interest rate of approximately 4.02% per quarter to achieve the desired growth. Yes, you heard it right, 4.02% every three months! Quite the acrobatic interest rate, wouldn't you say?

*applause and confetti*

But remember, my calculations are all in good fun! For an accurate assessment, I suggest consulting a financial professional who doesn't rely on juggling routines and balloon animals. Keep investing and keep smiling! πŸŽͺ

To find the rate of interest, we need to use the formula for compound interest:

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

Where:
A = Final amount
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times the interest is compounded per year
t = Number of years

We are given:
P = $900
A = $1400
n = 4 (compounded quarterly)
t = 12 years

Substituting the given values into the formula, we have:

$1400 = $900(1 + r/4)^(4*12)

Simplifying further:

$1400/$900 = (1 + r/4)^48

Divide 900 on both sides:

1.5556 = (1 + r/4)^48

Taking the 48th root on both sides:

(1.5556)^(1/48) = 1 + r/4

(1.5556)^(1/48) - 1 = r/4

r/4 = (1.5556)^(1/48) - 1

r = 4 * [(1.5556)^(1/48) - 1]

Using a calculator, we can evaluate the right-hand side to find the value of r:

r β‰ˆ 4 * (0.0163)

r β‰ˆ 0.065

Therefore, the interest rate is approximately 0.065, or 6.5% when compounded quarterly.

To determine the rate of interest, we can use the formula for compound interest:

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

where:
A = the final amount ($1400 in this case)
P = the principal amount ($900 in this case)
r = the interest rate (unknown)
n = the number of times the interest is compounded per year (quarterly compounding in this case, so n = 4)
t = the time in years (12 years in this case)

First, we need to rewrite the formula to solve for r:

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

Substituting the given values:

r = ( (1400/900)^(1 / (4*12)) ) - 1

Now, let's calculate this:

r = (1.5556^(1 / 48)) - 1

Using a calculator, we find:

r β‰ˆ 0.0123

So, the interest rate required for the investment to grow from $900 to $1400 in 12 years with quarterly compounding is approximately 1.23% per quarter.