Question

A sum of money under compound interest doubles itself in 4 years. In how many years will it become 16 times itself?
\[(A)\text{ 12}\]
\[(B)\text{ 16}\]
\[(C)\text{ 8}\]
\[(D)\text{ }6\]

Answer 1

Hint: Here, we have to find the time period after which the principal amount becomes 16 times itself meaning the total amount after adding compound interest becomes 16 times 16 times the principal amount. We are given that after 4 years, amount becomes double. We will use formula for calculating total amount which is given by \[A=P{{(1+\dfrac{R}{100})}^{T}}\]where A is total amount with compound interest, P is principal amount, R is rate of interest per annum and T is the time period.

Complete step by step answer:
We are given that the sum of money (principal amount) becomes double after 4 years at some rate of interest. Here, we are neither given a principal amount nor rate of interest. Hence, let us suppose the principal amount as x and rate of interest as r. As we know, total amount is calculated by the following formula \[A=P{{(1+\dfrac{R}{100})}^{T}}\] where P is the principal amount, R is rate of interest and T is time period. Putting values in the formula, we get,
\[A=x{{(1+\dfrac{r}{100})}^{4}}\]
Since the amount becomes double after 4 years, then A becomes 2x.
Therefore,A
  $2x=x{{(1+\dfrac{r}{100})}^{4}}$
eliminating x from both sides, we get
\[2={{(1+\dfrac{r}{100})}^{4}}\]………………equation (1)

Now for being 16 times we can write-
\[16x={{x(1+\dfrac{r}{100})}^{T}}\]
Eliminating x from both sides, we get
\[16={{(1+\dfrac{r}{100})}^{T}}\]
Since 16 is four times the power of 2, we can write 16 as \[{{(2)}^{4}}\].
Therefore,
\[{{(2)}^{4}}={{(1+\dfrac{r}{100})}^{T}}\]
From equation (1),
\[{{({{(1+\dfrac{r}{100})}^{4}})}^{4}}={{(1+\dfrac{\mu }{100})}^{T}}\]
\[{{(1+\dfrac{\mu }{100})}^{16}}={{(1+\dfrac{\mu }{100})}^{T}}\]
Comparing both, we conclude that T=16.
Time required is 16 years.

So, the correct answer is “Option B”.

Note: students should take care in putting variable values in formula while computing. It must also be noted that the original amount becomes 16 times and not the amount becomes should not solve equation (1) to find value of r to ease the calculations.