Question

At what rate percent (simple interest) will a sum of money double itself in 16 years ?
\[
  A.{\text{ }}6\dfrac{1}{4}\% \\
  B.{\text{ }}5\dfrac{1}{4}\% \\
  C.{\text{ }}5\dfrac{3}{4}\% \\
  D.{\text{ }}6\dfrac{3}{4}\% \\
\]

Answer 1

Hint: Let us use simple interest formula \[\dfrac{{{\text{P*R*T}}}}{{{\text{100}}}}\] to find interest if P is the principal amount, R is rate of interest per annum and T is the time period.

Complete Step-by-Step solution:

Let the rate of interest per annum will be r%
The starting principal amount will be equal to P.
Then according to the question amount after 16 years will be equal to 2P.
As we know that the amount after 16 years will be the sum of the initial amount (P) and the interest of 16 years.
So, 2P = P + simple interest for 16 years.
So, simple interest for 16 years = 2P – P = P (1)
As we know that the interest in T years if P is the principal amount and R is the rate of interest per annum is given by S.I. = \[\dfrac{{{\text{P*R*T}}}}{{{\text{100}}}}\].
So, total interest of amount P after 16 years will be \[\dfrac{{{\text{P*r*16}}}}{{{\text{100}}}}\] = \[\dfrac{{16\Pr }}{{100}}\]
So, using equation 1 we can write.
P = \[\dfrac{{16\Pr }}{{100}}\]
Now dividing both sides of the above equation by \[\dfrac{{100}}{{16{\text{P}}}}\]. We get,
r = \[\dfrac{{100}}{{16}}\] = \[\dfrac{{25}}{4}\] = \[6\dfrac{1}{4}\]
So, the rate of interest to double the amount in 16 interests will be equal to \[6\dfrac{1}{4}\]%.
Hence, the correct option will be A.

Note: Whenever we come up with this type of problem then first, we have to assume the principal amount as P. And the amount after 16 years as 2P. As, we know that amount after 16 years is the sum of initial amount and interest of 16 years. So, we put simple interest equal to 2P – P = P and then use a simple interest formula to find total interest after 16 years. After that we will solve the equation of simple interest to find the value of r (rate of interest).