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-1: X received 4/5 of what Y and Z together received. Statement-II: Y received 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?

C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

Reasoning : Logical Reasoning & Analytical Ability

Q 57 / 325

UPSC CSE Prelims 2024

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-1: X received 4/5 of what Y and Z together received.
Statement-II: Y received 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?

EXPLANATION

Correct Option

The question requires identifying which individual (X, Y, or Z) received the least amount. This necessitates determining the relative proportions of the amounts received by X, Y, and Z.

Analyzing Statement-I alone:

Statement-I provides the relationship: X = (4/5)(Y+Z).
Let the total amount be T = X+Y+Z. From the statement, X = (4/5)(T-X), which simplifies to 5X = 4T - 4X, leading to 9X = 4T, or X = (4/9)T.
This statement establishes X's share relative to the total but does not provide a relationship between Y and Z. Consequently, it is not possible to compare X, Y, and Z definitively to find the least amount. For example, if X = 4 units, then Y+Z = 5 units. Y could be 1 and Z could be 4, making Y the least. Alternatively, Y could be 4 and Z could be 1, making Z the least. Thus, Statement-I alone is insufficient.

Analyzing Statement-II alone:

Statement-II provides the relationship: Y = (2/7)(X+Z).
Similarly, from the total amount T = X+Y+Z, Y = (2/7)(T-Y), which simplifies to 7Y = 2T - 2Y, leading to 9Y = 2T, or Y = (2/9)T.
This statement establishes Y's share relative to the total but does not provide a relationship between X and Z. Therefore, it is not possible to compare X, Y, and Z definitively to find the least amount. Thus, Statement-II alone is insufficient.

Analyzing both Statements together:

We have two equations:

X = (4/5)(Y+Z)
Y = (2/7)(X+Z)

Substitute the expression for X from equation (1) into equation (2):

Y = (2/7) [ (4/5)(Y+Z) + Z ]

Y = (2/7) [ (4Y/5) + (4Z/5) + Z ]

Y = (2/7) [ (4Y/5) + (9Z/5) ]

Y = (8Y/35) + (18Z/35)

Multiply the entire equation by 35 to eliminate denominators:

35Y = 8Y + 18Z

27Y = 18Z

Y = (18/27)Z

Y = (2/3)Z

Now, substitute Y = (2/3)Z back into equation (1):

X = (4/5) [ (2/3)Z + Z ]

X = (4/5) [ (2Z+3Z)/3 ]

X = (4/5) [ (5Z)/3 ]

X = (4Z)/3

Thus, the amounts received by X, Y, and Z are in the following proportions:

X = (4/3)Z
Y = (2/3)Z
Z = Z

To compare these values, let Z = 3 units (to work with integers). Then:

Y = (2/3)*3 = 2 units
X = (4/3)*3 = 4 units

The amounts are in the ratio X:Y:Z = 4:2:3. From this ratio, Y received the least amount (2 units). Therefore, both statements together are sufficient to answer the question.

Incorrect Options:

Option (1) is incorrect because neither Statement-I nor Statement-II alone provides enough information to determine the least amount received, as demonstrated in the individual analysis of each statement.

Option (2) is incorrect because neither Statement-I nor Statement-II alone is sufficient to answer the question. Both statements independently leave the relative amounts of two variables undetermined.

Option (4) is incorrect because, as shown in the combined analysis, using both statements together allows for the determination of the exact ratios of amounts received by X, Y, and Z, thereby identifying the individual who received the least amount.