Question

A sum of money lent out at C.I at a certain rate per annum doubles itself in 5 years. Find in how many years will the money become eight times of itself at the same rate of interest p.a.

Answer 1

Hint: Here, we have to find the time period after which principal amount becomes 8 times itself meaning total amount becomes 8 times principal amount. We are given information for solving these types of questions such as the time after which the 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, P is principal amount, R is rate of interest and T is time period.

Complete step by step answer:
We are given that the sum of money (principal amount) becomes double after 5 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 principal amount, R is rate of interest and T is time period. Putting value in formula, we get
\[\Rightarrow A=P{{(1+\dfrac{r}{100})}^{5}}\]
Since amount becomes double the principal amount after 5 years, then A becomes 2x
Therefore,
\[\Rightarrow 2x=x{{(1+\dfrac{r}{100})}^{5}}\]
Eliminating x from both sides, we get
\[\Rightarrow 2={{(1+\dfrac{r}{100})}^{5}}\]………………. equation (1)
Now we have to find T when the amount becomes 8 times the principal amount. Therefore, if the principal amount is equal to x, then amount becomes 8x.
Using these values in formula \[A=P{{(1+\dfrac{R}{100})}^{T}}\], we get
\[\Rightarrow 8x=x{{(1+\dfrac{r}{100})}^{T}}\]
Eliminating x from both sides, we get
\[\Rightarrow 8={{(1+\dfrac{r }{100})}^{T}}\]

Since 8 is cube of 2, we can write 8 as \[{{(2)}^{3}}\],
Therefore,
\[\Rightarrow {{(2)}^{3}}={{(1+\dfrac{r }{100})}^{T}}\]
From equation (1),
\[\Rightarrow {{[{{(1+\dfrac{r}{100})}^{5}}]}^{3}}={{(1+\dfrac{r }{100})}^{T}}\]
\[\Rightarrow {{(1+\dfrac{r}{100})}^{15}}={{(1+\dfrac{r }{100})}^{T}}\]
Comparing both, we conclude that T=15.
Time required is 15 years.

So, the correct answer is “15 Years”.

Note: students should take care in putting variable values in the formula while computing. It must also be noted that the original amount becomes 8 times and not the amount after 5 years. Students can make this question long by solving equation (1) properly which is not required.