A sum of money would double itself in 10 years. The number of years it would be four times is (assume compound interest):
what a lot of useless and erroneous calculation!
it doubles in 10 years.
so it would double again (quadruple) in 10 more years, or 20 in all.
Apologies for the mistake in the calculation. You are correct. If the sum of money doubles in 10 years, it would double again (quadruple) in another 10 years, making it a total of 20 years. Thank you for catching that error.
To find the number of years it would take for a sum of money to quadruple itself, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal investment (initial amount of money)
r = annual interest rate (expressed as a decimal)
n = number of times that interest is compounded per year
t = number of years
Given that the sum of money doubles in 10 years, we can set up the equation as:
2P = P(1 + r/n)^(nt)
Simplifying this equation, we can cancel out P from both sides:
2 = (1 + r/n)^(nt)
Now we need to determine the value of (nt) to quadruple the initial investment. Substituting 4 for the future value (A) in the equation, we have:
4 = (1 + r/n)^(nt)
To find the value of (nt), we can take the logarithm of both sides of the equation. Let's use the natural logarithm (ln):
ln(4) = ln((1 + r/n)^(nt))
Using the laws of logarithms, we can bring down the exponent (nt) and solve for it:
nt * ln(1 + r/n) = ln(4)
Finally, solving for (nt) using the values given would give us the number of years it would take for the sum of money to quadruple itself.
To find the number of years it would take for the sum of money to quadruple, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (4 times the original amount)
P = the principal (the original amount)
r = the annual interest rate (unknown)
n = the number of times interest is compounded per year (unknown)
t = the time in years (unknown)
We know that the sum of money doubles in 10 years, so A/P = 2. Plugging this into the formula, we get:
2 = (1 + r/n)^(10n)
To find the number of years it would take for the sum to quadruple, we want to find when A/P = 4. Plugging this into the formula, we get:
4 = (1 + r/n)^(10n)
We need to solve for n to find the number of years. We can do this by taking the logarithm of both sides:
log(4) = log[(1 + r/n)^(10n)]
Using the power rule of logarithms, we can bring the exponent down:
log(4) = 10n * log(1 + r/n)
Rearranging the equation, we get:
log(1 + r/n) = log(4) / (10n)
Now we can solve for n. Let's assume that the annual interest rate is 100%, meaning r = 1 (the interest rate is 100% higher than the principal). Plugging this in, we get:
log(1 + 1/n) = log(4) / (10n)
We can graph this equation to find the value of n where the line intersects with log(4) / (10n) = 1:
In this case, it looks like the line intersects with log(4) / (10n) = 1 when n is approximately 6.
So, the number of years it would take for the sum of money to quadruple is around 6 years.