Question 874076: Isabella invested $500 at 6% annual interest, compounded quarterly. The value, A, of an investment can be calculated using the equation A=P(1+r/n)^nt, where P is the initial investment, r is the interest rate, n is the number of times the interest is compounded each year, and t is time in years. Exactly how long will it take for her investment to be worth four times as much (quadruple)in value?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! A = P * (1 + (r/n))^(n*t)
replace what you know in the formula and solve for what you don't know.
what you know:
r = 6% divided by 100% = .06 because r is always the decimal equivalent of the % which is always the percent divided by 100%.
n = 4 because there are 4 compounding periods per year (quarterly compounding).
r/n = .06 / 4 = .015
P = $500.
the formula is, once again:
A = P * (1+(r/n)^(n*t
you are given that A is going to be a quadruple of P which means that A will be equal to 4 * $500, so A is equal to $2000.
to summarize what you know:
n = 4
r = .06
r/n = .015
P = $500
A = $2000
t = unknown since that what you need to solve for.
your formula of:
A = P * (1 + (r/n))^(n*t) becomes:
$2000 = $500 * (1 + .015)^4t which becomes:
$2000 = $500 * (1.015)^4t
divide both sides of this formula by $500 to get:
$2000 / $500 = (1.015)^4t
simplify to get:
4 = (1.015)^4t
since the unknown is in the exponent, this calls for using logarithms to solve for it.
take the log of both sides of this equation to get:
log(4) = log((1.015)^4t)
use law of logarithms formula number 1 (shown below the final answer) to convert this equation to:
log(4) = 4t * log(1.015)
divide both sides of this formula by log(1.015) to get:
log(4) / log(1.015) = 4t
use your calculator to solve for 4t to get:
4t = 93.11105126
divide both sides of this equation by 4 to get:
t = 23.27776282
that's your final answer.
logarithm law number 1 is equal to:
log(x^a) = a * log(x)
in your problem, a was equal to 4t and x was equal to 1.015.
that's how you got log(1.015^4t) = 4t*log(1.015)
to confirm your answer is correct, substitute for t in the original equation to see if that equation becomes true.
the original equation is:
$2000 = $500 * (1.015)^4t
replace t with 23.27776282 to get:
$2000 = $500 * (1.015)^(4 * 23.27776282)
simplify the right side of this equation to get:
$2000 = $2000.
This confirms the solution is good.
the solution is:
t = 23.27776282 which you can round off to 23.28 years depending on how accurate you need the solution to be.
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