Question : Let XYZ be a 3-digit number and the difference between XYZ and ZYX is equal to PQR. Is (P + R) equal to Q ?
Statement-I : P = 3
Statement-II : R = 6
Which one of the following is correct in respect of the above Question and the Statements ?
Analyze XYZ - ZYX Difference to Check P+R = Q
Let the 3-digit number XYZ be represented algebraically as: \(XYZ = 100X + 10Y + Z\) Let the 3-digit number ZYX be represented algebraically as: \(ZYX = 100Z + 10Y + X\) For XYZ and ZYX to be 3-digit numbers, \(X \neq 0\) and \(Z \neq 0\). Assuming \(XYZ > ZYX\), we must have \(X > Z\). Thus, \(X > Z \ge 1\). The difference \(X - Z\) is a positive integer between 1 and 8.
Calculate the Difference XYZ - ZYX
The difference is calculated as:
\((XYZ - ZYX) = (100X + 10Y + Z) - (100Z + 10Y + X)\) \(= 99X - 99Z\) \(= 99(X - Z)\)This difference equals PQR, represented as \(100P + 10Q + R\). Therefore, \(100P + 10Q + R = 99(X - Z)\).
Determine the Relationship between P, Q, and R
Let's examine the structure of \(99(X - Z)\):
- If \(X - Z = 1\), \(99 \times 1 = 99\). PQR = 099 (\(P=0, Q=9, R=9\)). \(P+R = 0+9 = 9 = Q\).
- If \(X - Z = 2\), \(99 \times 2 = 198\). PQR = 198 (\(P=1, Q=9, R=8\)). \(P+R = 1+8 = 9 = Q\).
- If \(X - Z = 3\), \(99 \times 3 = 297\). PQR = 297 (\(P=2, Q=9, R=7\)). \(P+R = 2+7 = 9 = Q\).
- If \(X - Z = 4\), \(99 \times 4 = 396\). PQR = 396 (\(P=3, Q=9, R=6\)). \(P+R = 3+6 = 9 = Q\).
- If \(X - Z = 5\), \(99 \times 5 = 495\). PQR = 495 (\(P=4, Q=9, R=5\)). \(P+R = 4+5 = 9 = Q\).
- If \(X - Z = 6\), \(99 \times 6 = 594\). PQR = 594 (\(P=5, Q=9, R=4\)). \(P+R = 5+4 = 9 = Q\).
- If \(X - Z = 7\), \(99 \times 7 = 693\). PQR = 693 (\(P=6, Q=9, R=3\)). \(P+R = 6+3 = 9 = Q\).
- If \(X - Z = 8\), \(99 \times 8 = 792\). PQR = 792 (\(P=7, Q=9, R=2\)). \(P+R = 7+2 = 9 = Q\).
In all cases, the tens digit \(Q\) is 9, and the sum of the hundreds digit \(P\) and the units digit \(R\) is also 9 (\(P+R=9\)). Therefore, \(P + R = Q\) is always true.
Conclusion on Statements
The question "Is (P + R) equal to Q?" can be answered affirmatively using only the general properties of the number system, specifically the result \(XYZ - ZYX = 99(X - Z)\), which leads to \(P+R=Q\). The specific values \(P=3\) (Statement-I) and \(R=6\) (Statement-II) are not required to arrive at this conclusion.
Hence, the question can be answered even without using both statements.
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