Alison deposits $500 into a new savings account that earns 5 percent interest compounded annually. If Alison makes no additional deposits or withdrawals, how many years will it take for the amount in the account to double?
the answer is 15 but i think its 14. please explain.
No Dublin, that is for compounded continuously. This problem states anually.
Sra is right.
The "calculator" on that webpage is using this formula
Amount = Principal(1+i)^n
we have 1000=500(1.05)^n
2 = 1.05^n
take log of both sides
log2 = log(1.05)^n
log2 = nlog1.05
n = log2/log1.05 = 14.2
after 14 years, your money has not yet doubled, close, but not yet.
Amount = 500(1.05)^14 = 989.97
So I guess they are right at 15, since you have to go into the 15th year to double your deposit.
I don't think so because A = Pe^rt
1000 = 500e^0.05 * t
2 = e^(0.05 * t)
ln2 = 0.05t
t=ln2 / 0.05
t= 13.86
You have to round up so 14 is the correct answer.
To calculate the number of years it takes for the amount in the account to double, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested for
In this case, Alison deposits $500 into the savings account, and the interest rate is 5 percent (or 0.05) compounded annually. We need to find the number of years it will take for the amount to double.
Let's plug in the values into the formula:
1000 = 500(1 + 0.05/1)^(1t)
Now we need to solve for t. We can simplify the equation further:
2 = (1.05)^t
To solve for t, we need to take the logarithm of both sides of the equation. In this case, we can use the natural logarithm (ln) or logarithm base 10 (log):
ln(2) = t * ln(1.05)
or
log(2) = t * log(1.05)
Using a calculator, we can find the value of t:
t = ln(2) / ln(1.05)
or
t = log(2) / log(1.05)
Calculating either of these equations, we find that t ≈ 14.21 (using ln) or t ≈ 14.45 (using log). Hence, it will take approximately 14.21 or 14.45 years for the amount in the account to double.
Therefore, the answer is not precisely 14 years, as you mentioned, but somewhere between 14 and 15 years. Since the question asks for the number of whole years, the closest whole number that fits is 15.