at what rate converted every 3 months will a sum of money triple itself in 18 years and 6 months.
or, using the compound interest formula
(1 + i)^74 = 3 , where i is the quarterly rate
take 74th root of both sides
1+i = 1.01495..
i = .01495..
annual rate compounded quarterly
= 4(.01595..)
= .0598 or 5.98%
I assume you mean compounded
r = interest rate specified usually per year /4 because compounding every quarter year
18.5 years * 4 = 74 periods
3 = x^74
log 3 = 74 log x
log x = .00644758
x =1.015
so
1.5 percent per period or r = 4*1.5 = 6%
To find the rate at which a sum of money will triple itself in a given time period, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Final amount
P = Initial principal (sum of money)
r = Interest rate (annual rate as a decimal)
n = Number of times interest is compounded per year
t = Number of years
In this case, we need to find the interest rate that will cause the sum of money to triple itself in 18 years and 6 months, which is equivalent to 18.5 years.
Since the money needs to triple, the final amount is three times the initial principal, so A = 3P.
Plugging in the values into the formula, we have:
3P = P(1 + r/n)^(n * 18.5)
Divide both sides by P to simplify:
3 = (1 + r/n)^(n * 18.5)
To proceed further, we need to make an assumption about the compounding frequency (n) since the question does not provide this information. Let's assume the compounding is done quarterly, i.e., n = 4 (every 3 months).
Now we have:
3 = (1 + r/4)^(4 * 18.5)
To solve for r, we need to isolate it. Take the natural logarithm (ln) of both sides:
ln(3) = ln((1 + r/4)^(4 * 18.5))
Using the logarithmic property ln(a^b) = b * ln(a), the equation becomes:
ln(3) = (4 * 18.5) * ln(1 + r/4)
Now, divide both sides by (4 * 18.5) and solve for ln(1 + r/4):
ln(3)/(4 * 18.5) = ln(1 + r/4)
Finally, raise both sides as a power of e (exponential function) and solve for r:
e^(ln(3)/(4 * 18.5)) = 1 + r/4
Subtract 1 from both sides and multiply by 4 to solve for r:
r = 4 * (e^(ln(3)/(4 * 18.5)) - 1)
Calculating this expression will give you the rate at which the sum of money needs to be converted every 3 months to triple itself in 18 years and 6 months.
To find the rate at which a sum of money will triple itself in a specified time period, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A is the future value of the investment
P is the principal amount (initial sum of money)
r is the annual interest rate (in decimal form)
n is the number of times interest is compounded per year
t is the time period in years
In this case, we want the future value (A) to be three times the principal (P), and the time period (t) is given as 18 years and 6 months.
Let's start by converting the time period to years. There are 12 months in a year, so 6 months is equivalent to 1/2 year.
Therefore, the time period (t) = 18 + 6/12 = 18.5 years.
Now, we can plug in the values into the formula:
3P = P(1 + r/n)^(nt)
Dividing both sides of the equation by P:
3 = (1 + r/n)^(nt)
Substituting the values:
3 = (1 + r/n)^(18.5n)
To solve this equation for the interest rate (r/n), we need to use numerical methods or trial and error. However, we can make an educated guess based on the given information.
Typically, interest is compounded on a quarterly basis (n = 4) for every 3 months. So, we can start by assuming n = 4 and find the interest rate (r) that satisfies the equation.
Let's use an online calculator or a spreadsheet to iterate and find the value of r that makes the equation true.
After finding the value for r, we can multiply it by 4 to get the annual interest rate in decimal form, which will give us the rate at which the sum of money needs to be converted every 3 months to triple itself in 18 years and 6 months.