Question

The CI on a certain sum for $2$ years at $10\% $ per annum is \[{\text{Rs}}{\text{.}}\;525\]. The SI on the same sum for double the time at half the rate percent per annum is:
A.\[{\text{Rs}}{\text{.}}\;400\]
B.\[{\text{Rs}}{\text{.}}\;500\]
C.\[{\text{Rs}}{\text{.}}\;600\]
D.\[{\text{Rs}}{\text{.}}\;800\]

Answer 1

Hint: We know that the difference of amount and principal is known as the compound interest. So, we use the formula of amount to find the principal amount compounded yearly when compound interest is \[{\text{Rs}}{\text{.}}\;525\] and then apply the formula of simple interest.

Formula used: - The formula of simple interest with principal $P$, rate $r$ and time $t$ is given by,
\[{\text{S}}{\text{.I}}. = \dfrac{{P \times r \times t}}{{100}}\]
The formula of amount with principal $P$, rate $r$ and time $t$ is given by,
$A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$

Complete step-by-step answer:
We know that the CI is \[{\text{Rs}}{\text{.}}\;525\] on a certain sum for $2$ years at $10\% $ per annum.
We know that the C.I. is the difference of amount and principal. So,
\[{\text{CI}} = A - P\]
${\text{CI}} = P{\left( {1 + \dfrac{r}{{100}}} \right)^t} - P$
Substitute all the values in the above formula to find the principal amount.
$ \Rightarrow 525 = P{\left( {1 + \dfrac{{10}}{{100}}} \right)^2} - P$
$ \Rightarrow 525 = P\left[ {{{\left( {1 + \dfrac{1}{{10}}} \right)}^2} - 1} \right]$
$ \Rightarrow 525 = P\left[ {{{\left( {\dfrac{{11}}{{10}}} \right)}^2} - 1} \right]$
$ \Rightarrow 525 = P\left[ {\dfrac{{121}}{{100}} - 1} \right]$
$ \Rightarrow 525 = P\left[ {\dfrac{{21}}{{100}}} \right]$
$ \Rightarrow 52500 = 21 \times P$
$ \Rightarrow P = 2500$
 The principal amount or the sum is \[{\text{Rs}}{\text{.}}\;2500\] for the given CI.
Now, we need to find SI for the same sum if time is doubled at half the rate percent per annum.
If time is doubled then we get,
$ \Rightarrow t = 2 \times 2\;{\text{years}}$
$ \Rightarrow t = 4\;{\text{years}}$
The half of the given rate percent per annum is:
$ \Rightarrow r = \dfrac{{10\% }}{2}$
$ \Rightarrow r = 5\% $
 Substitute all the values in the formula of simple interest.
\[ \Rightarrow {\text{S}}{\text{.I}}. = \dfrac{{2500 \times 5 \times 4}}{{100}}\]
\[ \Rightarrow {\text{S}}{\text{.I}}. = 25 \times 5 \times 4\]
\[ \Rightarrow {\text{S}}{\text{.I}}. = 500\]
 Therefore, for the sum of \[{\text{Rs}}{\text{.}}\;2500\] the required simple interest is \[{\text{Rs}}{\text{.}}\;500\].
So, option (B) is the correct answer.

Note: We need to find the principal or sum to find the simple interest. Do not consider the amount as the sum otherwise this may lead to the incorrect answer. Also, make sure that the CI is the difference of amount and principal instead of the addition.