Question

In what time will the interest on a certain sum of money at 6% be $\dfrac{5}{8}$of itself?
A) 9 years and 4 months
B) 10 years and 5 months
C) 10 years and 4 months
D) 9 years and 5 months

Answer 1

Hint: Simple interest or interest figured on the principal amount only, for the duration of the loan.
The rate % (percentage) per annum on the principal amount and the duration of the loan are the factors for calculating simple interest.

Complete step by step solution:
Step 1
Let the time period = t years
Let the principle amount = P rupees
Given:
Rate % per annum on principal amount P = 6%

Step 2
Condition for interest stated in question:
Interest amount is $\dfrac{5}{8}$ times of principal amount, P.
$ \Rightarrow {\rm{I}} = \dfrac{5}{8}{\rm{P}}$ …… (1)

Step 3
Here, I in terms of P, P and r are given, only one unknown time period, t
Therefore,
Using simple interest formula, where
                                                              ${\rm{I}} = \dfrac{{\Pr t}}{{100}}$
$\begin{array}{l}
 \Rightarrow {\rm{ }}\dfrac{5}{8}P = \dfrac{{P \times 6 \times t}}{{100}}\\
 \Rightarrow {\rm{ }}\dfrac{5}{8}{P} = \dfrac{{{P} \times 6 \times t}}{{100}}\\
 \Rightarrow {\rm{ t}} = \dfrac{{5 \times 100}}{{8 \times 6}}\\
 \Rightarrow {\rm{ t}} = \dfrac{{125}}{{12}}
\end{array}$
Hence, t = 10.41 years

Step 4
To convert the time period, t according to the options:
We know, 1 year = 12 months
          $\begin{array}{l}
0.41{\rm{ years}} = 0.41 \times 12{\rm{ months}}\\
{\rm{ = 4}}{\rm{.92 months}}\\
{\rm{ }} \simeq 5{\rm{ months}}
\end{array}$
Therefore, time period = 10 years and 5 months

In 10 years and 5 months the interest on a certain sum of money at 6% will be $\dfrac{5}{8}$of itself. Thus, the correct option is (B).

Note:
When numbers of years are converted into days, it is always multiplied by 365, whether it is a leap year or an ordinary year.
Day on which the money is borrowed is not counted but the day on which the money is returned is counted.