Please Help.
How long, to the nearest tenth of a year, will it take $12,500 to grow to $20,000 at 6.5% annual interest compounded quartely? (Use the formula for compound interest with n compoundings per year to solve for t.)
With k compounding a year, the compound interest formula becomes:
FV = PV*Rkn
where
FV=future value
PV=present value
r=annual rate of interest, in fraction.
For example, 0.12 stands for 12%.
k=number of compounding a year, 4 for compounding every three months.
n=number of years
R=compounding rate, = 1+r/k
For example,
at 8% annual interest compounded 4 times a year, $10000 will accumulate to $20000 in n years.
20000=10000*(1+0.08/4)4n
divide by 10000,
1.024n = 2.0
take log on both sides
4n log(1.02) = log(2.0)
n = (1/4)log(2)/log(1.02)
=8.75 years
Still not sure what the answer is.
To find the time it will take for an amount to grow with compound interest, we can use the formula for compound interest, which is:
A = P(1 + r/n)^(nt)
Where:
A = the amount after t years
P = the principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = time in years
In this case, we have:
P = $12,500
A = $20,000
r = 6.5% = 0.065 (since it's a decimal)
n = 4 (quarterly compounding)
We need to solve for t.
Substituting these values into the formula, we get:
$20,000 = $12,500(1 + 0.065/4)^(4t)
Now, let's solve for t:
Divide both sides of the equation by $12,500:
$20,000 / $12,500 = (1 + 0.065/4)^(4t)
Simplify both sides:
1.6 = (1 + 0.01625)^(4t)
Now, take the natural logarithm (ln) of both sides of the equation to isolate the exponent:
ln(1.6) = ln((1.01625)^(4t))
Using the logarithmic property ln(a^b) = b*ln(a), we can bring the exponent down:
ln(1.6) = 4t*ln(1.01625)
Divide both sides by 4ln(1.01625):
t = ln(1.6) / (4ln(1.01625))
Using a calculator, evaluate the right-hand side:
t ≈ 4.2 years (rounded to the nearest tenth of a year)
Therefore, it will take approximately 4.2 years (or 4 years and 2 months) for $12,500 to grow to $20,000 at 6.5% annual interest compounded quarterly.