Theresa invested $5,000 in an account she expects will earn 7% annually. Approximately how many years will it take for the account to double in value?
solve:
5000(1.07)^n = 10000
1.07^n = 2
log both sides
log 1.07^n = log 2
n log1.07 = log2
n = log2/log1.07 = appr 10.24
It will take appr 10 years
To determine how many years it will take for the account to double in value, we need to use the concept of compound interest. The formula to calculate the future value of an investment with compound interest is:
FV = PV * (1 + r)^n
Where:
FV is the future value of the investment,
PV is the present value of the investment,
r is the interest rate, and
n is the number of compounding periods.
In this case, the present value (PV) is $5,000, the interest rate (r) is 7% or 0.07, and we want to find the number of years (n) when the account doubles, so the future value (FV) will be $10,000.
Plugging these values into the formula, we get:
$10,000 = $5,000 * (1 + 0.07)^n
Now, we need to solve for n. To do that, we can divide both sides of the equation by $5,000:
2 = (1 + 0.07)^n
To isolate n, we can take the logarithm of both sides of the equation:
log(2) = log((1 + 0.07)^n)
Using logarithmic properties, we can bring the exponent n down:
log(2) = n * log(1 + 0.07)
Finally, we divide both sides of the equation by log(1 + 0.07) to solve for n:
n = log(2) / log(1 + 0.07)
Using a calculator, we can evaluate this expression to find:
n ≈ 10.24
Therefore, it will take approximately 10.24 years for Theresa's account to double in value.
To find out approximately how many years it will take for the account to double in value, we can use the formula for compound interest.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years
In this case, the principal amount (P) is $5,000, the annual interest rate (r) is 7% (or 0.07 as a decimal), and we want to find the number of years it takes for the account to double, so the final amount (A) is $10,000.
Plugging in these values, the formula becomes:
$10,000 = $5,000(1 + 0.07/n)^(n*t)
To simplify the calculation and get an approximate value, we can assume that the interest is compounded annually (n = 1). Therefore, the formula becomes:
$10,000 = $5,000(1 + 0.07)^(1*t)
To find the value of t, we can rearrange the equation:
(1 + 0.07)^t = 10,000 / 5,000
1.07^t = 2
Now, we can solve for t by taking the logarithm of both sides:
t * log(1.07) = log(2)
t ≈ log(2) / log(1.07)
Using a calculator, we can find the approximate value of t as:
t ≈ 10.24
Therefore, it will take approximately 10.24 years for the account to double in value.