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 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?
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- (b) The Question can be answered by using either Statement alone
- (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone ✓ UPSC's answer
- (d) The Question cannot be answered even by using both the Statements together
Why the answer is (c)
• Let the total amount be T, so X + Y + Z = T.
• From Statement I: X = (4/5)(Y + Z). Since Y + Z = T - X, we get X = (4/5)(T - X), which simplifies to 5X = 4T - 4X, or 9X = 4T, so X = (4/9)T.
• From Statement II: Y = (2/7)(X + Z). Since X + Z = T - Y, we get Y = (2/7)(T - Y), which simplifies to 7Y = 2T - 2Y, or 9Y = 2T, so Y = (2/9)T.
• Neither statement alone provides the value of Z, so the least amount cannot be determined from either statement individually.
• Using both statements together, Z = T - X - Y = T - (4/9)T - (2/9)T = (3/9)T.
• Comparing the shares: X = 4/9 T, Y = 2/9 T, Z = 3/9 T. Since 2/9 is the smallest fraction, Y received the least amount, making the question answerable only when both statements are used.
Why the other options are wrong
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered…
- Statement I alone only determines X's share relative to the total, leaving Y and Z's individual shares unknown, so the least amount cannot be identified.
- (b) The Question can be answered by using either Statement alone
- Neither statement alone is sufficient to determine the individual shares of all three persons, so the question cannot be answered by either statement alone.
- (d) The Question cannot be answered even by using both the Statements together
- Using both statements together allows the calculation of all three shares (X=4/9T, Y=2/9T, Z=3/9T), which clearly identifies Y as receiving the least amount.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.