if a sum of money doubles in 8 yrs, in how many years will it triple itself?Is the answer 16? If it is please tell me how to show the workings. I understood the concept but cannot express it.This is a simple interest sum.
No, in 16 years the original sum will quadruple.
Let N be the number of years required to triple.
The increase fraction per year (think of it as an annual interest rate) is r.
(1+r)^8 = 2
8 log(1+r) = log 2
log(1+r) = 0.03673
1+r = 1.0905
r = 0.0905
(1.0905)^N = 3
N = log3/log(1.0905)
= 12.68 years
To find out in how many years a sum of money will triple itself, we need to determine the time it takes for the sum to double first. Once we know that, we can simply double that time to obtain the time it takes for the sum to triple.
In this case, if a sum of money doubles in 8 years, we can use the concept of compound interest to find the time it takes for the principal amount to double. The formula for compound interest is:
A = P * (1 + r/n)^(n*t)
Where:
A = Final amount
P = Principal amount
r = Annual interest rate (expressed as a decimal)
n = Number of times interest is compounded per year
t = Time in years
In the given scenario, the sum doubles, so the final amount (A) is 2 times the principal amount (P). Based on this, we can set up the equation:
2P = P * (1 + r/n)^(n*8)
Now, we need to solve this equation for n. However, since it is a simple interest problem, we can assume that the interest is compounded annually (n = 1). Therefore, the equation becomes:
2P = P * (1 + r)^(8)
Simplifying further, we find:
2 = (1 + r)^8
Now, we need to solve for r. We can take the eighth root of both sides of the equation:
(1 + r) = 2^(1/8)
Now, subtracting 1 from both sides, we get:
r = 2^(1/8) - 1
Using a calculator, we find that r is approximately 0.098. This means the annual interest rate is 0.098 (or 9.8% as a percentage).
Now that we know the interest rate, we can find the time it takes for the sum to triple. Using the formula again, but this time with the final amount being 3 times the principal (A = 3P), we can solve for t:
3P = P * (1 + r)^(t)
Dividing both sides by P, we get:
3 = (1 + r)^t
Taking the logarithm (base 1 + r) of both sides, we can solve for t:
log base (1 + r) (3) = t
Using a calculator, we find that t is approximately 15.97. Rounded to the nearest whole number, the sum will triple itself in approximately 16 years.
Therefore, the answer is indeed 16 years.