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.

To find the value of $t$ (the time period for Rahul's investment at compound interest rate of $R\%$), we analyze both statements individually and then together.

Analysis of Statement (I):
Raj invested Rs. 10,000 in scheme A at a rate of $R\%$ simple interest for 3 years and received Rs. 3,000 as interest.
Using the simple interest formula:
$\mathrm{Simple\; Interest}=\frac{\mathrm{Principal}\times \mathrm{Rate}\times \mathrm{Time}}{100}$
Substituting the given values:
$3000=\frac{10000\times R\times 3}{100}$
$3000=300\times R$
$R=10$
Thus, Statement (I) gives us $R=10\%$. However, this statement provides no information about Rahul's investment or the time $t$. Therefore, Statement (I) alone is not sufficient to find $t$.

Analysis of Statement (II):
Rahul invested a certain amount (let it be $P$) at compound interest for $t$ years at the rate of $X\%$ per annum. The interest obtained is Rs. 1,900 less than the simple interest on the same principal $P$ at the same rate $X\%$ for 4 years.
This can be written mathematically as:
$P\times \frac{X\times 4}{100}-P\times [{(1+\frac{X}{100})}^t-1]=1900$
In this equation, the principal amount $P$, interest rate $X$, and time $t$ are all unknown variables. Therefore, Statement (II) alone is not sufficient to find $t$.

Combining Statement (I) and Statement (II):
Even if we combine both statements, we only know the value of $R=10\%$. Statement (II) introduces an unknown rate $X\%$ and an unknown principal $P$. Even if we assume $X=R=10\%$, the principal amount $P$ remains unknown in the equation, leaving us with one equation and two variables ($P$ and $t$). Thus, $t$ cannot be uniquely determined.
Consequently, both statements taken together are not sufficient to answer the question.