Question

A finance company declares that, with compound interest rate, a sum of money deposited by anyone will become 8 times in three years. If the same amount is deposited at the same compound-rate of interest, then in how many years it will become 128 times?
A. 4
B. 5
C. 6
D. 7

Answer 1

HINT: The formula for calculating the compound interest on any sum of money is given and thus the new amount that is what an investor gets finally, is
Now, the most important formula that is required to solve this question is as follows
 Amount \[=principle~\times {{(1+rate\%)}^{time}}\]

Complete step-by-step answer:

New amount is nothing but the principle when added with the interest that is gained in the given time frame.

As mentioned in the question, we have to find the time after which the same sum of money becomes 128 times.
Let the principle or the initial sum be x Rs.

For this, we would use the formula that is given in the hint which is as follows
In the first case,
It is given in the question as follows
Time=3 yrs, amount =8x, rate =R.
Now, on using the formula for the new amount, we get
\[\begin{align}
  & 8x = x~\times {{(1+R\%)}^{3}} \\
 & 8 = {{(1+R\%)}^{3}} \\
\end{align}\]
Now, on taking cube roots of both the side, we get
\[\begin{align}
  & (1+R\%)=2 \\
 & R\%=1 \\
\end{align}\]
Now as we know, the rate at which the principle is compounded upon or the rate of interest for the compound interest, we can find the solution.

Hence, in the second case, we have the information as mentioned in the question as follows
The new amount =128x, Time =n whereas the rate is the same which is R% that is 1.
So, by using the formula given in the hint, we can write as follows
\[\begin{align}
  & 128x=x~\times {{(1+1)}^{n}} \\
 & 128={{\left( 1+1 \right)}^{n}} \\
 & 128={{\left( 2 \right)}^{n}} \\
\end{align}\]
Now, on comparing the two sides, we get that
n=7
So, after 7 years, the amount will become 128 times the invested money.

Hence option (D) is the correct answer.

NOTE: The students can make an error if they don’t know the formula for calculating the amount which the principle becomes after being put under compound interest as follows
Amount \[=principle~\times {{(1+rate\%)}^{time}}\], because if they don’t know this above mentioned formula, then they cannot reach the final correct answer as it is the most important formula for solving this question.