To determine the rate of interest at which a sum of money will triple itself in 12 ½ years, we can use the compound interest formula. The formula is given as:
A = P(1 + r/n)^(nt)
Where:
A is the future value or the amount after time t,
P is the principal or the initial amount,
r is the interest rate (in decimal form),
n is the number of times interest is compounded per year, and
t is the time in years.
In this case, we want the amount to triple, so the future value (A) will be three times the principal (P). The time (t) is given as 12 ½ years, which can be expressed as 25/2 years.
Let's assume the principal is P and the interest rate is r.
Using the compound interest formula and substituting the given values, we have:
3P = P(1 + r/n)^(nt)
Simplifying the equation further, we get:
3 = (1 + r/n)^(nt)
Now, we can substitute the given values: n = 1 (compounded annually) and t = 25/2.
3 = (1 + r/1)^(1*(25/2))
3 = (1 + r)^(25/2)
To find the rate of interest (r), we need to isolate it in the equation. Taking the square root of both sides to get rid of the exponent, we have:
√3 = 1 + r
Subtracting 1 from both sides, we get:
√3 - 1 = r
Now we can calculate the approximate value of r using a calculator:
r ≈ √3 - 1
r ≈ 1.732 - 1
r ≈ 0.732
Therefore, the rate of interest required for a sum of money to triple itself in 12 ½ years is approximately 0.732, or 73.2% when expressed as a percentage.
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Step-by-step explanation:
To determine the rate of interest at which a sum of money will triple itself in 12 ½ years, we can use the compound interest formula. The formula is given as:
A = P(1 + r/n)^(nt)
Where:
A is the future value or the amount after time t,
P is the principal or the initial amount,
r is the interest rate (in decimal form),
n is the number of times interest is compounded per year, and
t is the time in years.
In this case, we want the amount to triple, so the future value (A) will be three times the principal (P). The time (t) is given as 12 ½ years, which can be expressed as 25/2 years.
Let's assume the principal is P and the interest rate is r.
Using the compound interest formula and substituting the given values, we have:
3P = P(1 + r/n)^(nt)
Simplifying the equation further, we get:
3 = (1 + r/n)^(nt)
Now, we can substitute the given values: n = 1 (compounded annually) and t = 25/2.
3 = (1 + r/1)^(1*(25/2))
3 = (1 + r)^(25/2)
To find the rate of interest (r), we need to isolate it in the equation. Taking the square root of both sides to get rid of the exponent, we have:
√3 = 1 + r
Subtracting 1 from both sides, we get:
√3 - 1 = r
Now we can calculate the approximate value of r using a calculator:
r ≈ √3 - 1
r ≈ 1.732 - 1
r ≈ 0.732
Therefore, the rate of interest required for a sum of money to triple itself in 12 ½ years is approximately 0.732, or 73.2% when expressed as a percentage.
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