A Question is given followed by two Statements I and II. Consider the Question and the Statements.

A certain amount was distributed among X, Y and Z.

Question: Who received the least amount?

Statement-I : X received $\frac{4}{5}$ of what Y and Z together received.

Statement-II : Y received $\frac{2}{7}$ of what X and Z together received.

Which one of the following is correct in respect of the above Question and the Statements?

The correct option is: The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

Let the amounts received by X, Y, and Z be represented by variables $x$, $y$, and $z$ respectively. The total amount distributed is $T=x+y+z$. We need to determine who received the least amount among X, Y, and Z.

Analyzing Statement-I:
According to Statement-I, X received $\frac{4}{5}$ of what Y and Z together received.
This can be written as:
$x=\frac{4}{5}(y+z)$
Multiplying both sides by 5:
$5x=4(y+z)$
Adding $4x$ to both sides to express it in terms of the total amount $T=x+y+z$:
$9x=4(x+y+z)=4T$
Hence, $x=\frac{4}{9}T\approx 0.44T$.
This leaves $y+z=\frac{5}{9}T\approx 0.56T$.
From this statement alone, we cannot determine the individual shares of Y and Z. For example, if Y receives a very small share and Z receives the rest, Y could be the least. If Z receives a very small share, Z could be the least. Thus, Statement-I alone is not sufficient.

Analyzing Statement-II:
According to Statement-II, Y received $\frac{2}{7}$ of what X and Z together received.
This can be written as:
$y=\frac{2}{7}(x+z)$
Multiplying both sides by 7:
$7y=2(x+z)$
Adding $2y$ to both sides to write in terms of the total amount $T$:
$9y=2(x+y+z)=2T$
Hence, $y=\frac{2}{9}T\approx 0.22T$.
This leaves $x+z=\frac{7}{9}T\approx 0.78T$.
From this statement alone, we cannot determine the individual shares of X and Z. Therefore, Statement-II alone is not sufficient.

Combining Statement-I and Statement-II:
When we combine both statements, we have:
1) $x=\frac{4}{9}T$
2) $y=\frac{2}{9}T$
Now we can easily find the share of Z:
$z=T-x-y$
$z=T-\frac{4}{9}T-\frac{2}{9}T=\frac{3}{9}T=\frac{1}{3}T$
Comparing the three shares in terms of $T$:
- X's share: $\frac{4}{9}T$ (or approx. 44.4%)
- Y's share: $\frac{2}{9}T$ (or approx. 22.2%)
- Z's share: $\frac{3}{9}T$ (or approx. 33.3%)
Since $\frac{2}{9}T<\frac{3}{9}T<\frac{4}{9}T$, Y received the least amount.

Therefore, we can answer the question uniquely by using both statements together, but not by using either statement alone.