Directions (61-63): The following questions are accompanied by two statements i.e. statement (I) and statement (II). You have to determine which statement (s) is/are sufficient/necessary to answer the questions.

Rahul invested in a bank at a rate of R% compound annually for t years. Find the value of t?
Statement (I): Raj invested Rs. 10,000 in scheme A at a rate of R% at a simple interest for 3 years then he gets Rs. 3000 as an interest.
Statement (II): If Rahul invested certain amount at a compound interest for t years at the rate of X%, then he gets Rs.1900 less interest than the interest he got when he invested same amount at simple interest at same rate for 4 years.

The correct option is: Neither statement (I) nor statement (II) by itself is sufficient to answer the question.

Let us analyze the question and the given statements step-by-step to determine their sufficiency for finding the value of $t$.

Given Information:
Rahul invested a certain sum of money (let it be $P$) in a bank at a rate of $R\%$ compound interest annually for $t$ years.
We need to find the value of $t$.

Analyzing Statement (I):
Statement (I) states: "Raj invested Rs. 10,000 in scheme A at a rate of R% at a simple interest for 3 years then he gets Rs. 3000 as an interest."
From this, we can calculate the interest rate $R$ using the simple interest formula:
$\mathrm{Simple\; Interest}=\frac{P\times R\times T}{100}$
Substituting the given values:
$3000=\frac{10000\times R\times 3}{100}$
$3000=300\times R$
$R=10\%$
Thus, Statement (I) gives us the value of $R=10$, but it provides no information about $t$ or the relationship between interest earned by Rahul. Therefore, Statement (I) alone is not sufficient.

Analyzing Statement (II):
Statement (II) states: "If Rahul invested certain amount at a compound interest for t years at the rate of X%, then he gets Rs.1900 less interest than the interest he got when he invested same amount at simple interest at same rate for 4 years."
Let the principal amount invested be $P_1$ and the rate be $X\%$.
The compound interest earned for $t$ years is:
${\mathrm{CI}}_t=P_1{(1+\frac{X}{100})}^t-P_1$
The simple interest earned for 4 years at the same rate $X\%$ is:
${\mathrm{SI}}_4=\frac{P_1\times X\times 4}{100}$
According to Statement (II):
${\mathrm{SI}}_4-{\mathrm{CI}}_t=1900$
Since the principal $P_1$ and the rate of interest $X$ are both unknown, and this rate $X$ is different from $R$, Statement (II) alone is not sufficient to find $t$.

Combining Statements (I) and (II):
Even if we combine both statements, Statement (I) gives us the value of $R$, but Statement (II) defines a completely different scenario involving an unknown rate of $X\%$ and an unknown principal $P_1$ instead of Rahul's original rate $R\%$. There is no connection established between $R$ and $X$ or the principal amounts, which leaves us with too many unknown variables ($P_1$, $X$, and $t$).
Therefore, even after combining both statements, we cannot find the value of $t$.

Hence, neither statement (I) nor statement (II) by itself is sufficient to answer the question.