PQ and QP represent two-digit numbers having P and Q as their digits.

it is evident that P, Q, R & S are single digit number

Statement 1: solve the addition problem in a conventional way to get-
PQ
+QP
------
RSR

maximum sum of two single digits can be 9+9=18

as Q+P=R so S=P+Q+1 (there has to be a carry forward of 1 as R & S are different digits)

and finally the hundreds place i.e R will be 1 (carry forward from the sum of P+Q)

so now we have R=1 & S=2 and P+Q=11 (because sum of P & Q has to yield a unit's digit 1 and maximum possible sum of any two single digit is 18)

Hence P+Q+R+S=11+1+2=14. Sufficient

Statement 2: no relation provided for the digits. Hence \(Insufficient\)

Option A

Hi All,

While this question might look complex, it's actually based on some basic arithmetic rules. If you don't recognize the "theory" behind this question, then you could still solve it with a bit of 'brute force' arithmetic and a little logic.

We're told that QP and PQ are two 2-digit numbers that have the same digits (just in reverse-order) and that RSR is a 3-digit number. We're asked for the value of P+Q+R+S.

1) PQ + QP = RSR.

With Fact 1, notice that the sum of the two 2-digit numbers is a 3-digit number. In this situation, the 3-digit number MUST begin with a 1 (there's no other possibility since the highest sum of two 2-digit numbers is 99+99 = 198). Thus, R = 1....

PQ + QP = 1S1

By extension, Q+P must "end" in a 1 AND PQ+QP must be large enough to create a 3-digit sum. From here, you can brute-force the options and see what happens....
P=2, Q=9... 29+92 = 121.... so S=2 and the answer to the question is 2+9+1+2 = 14
P=3, Q=8... 38+83 = 121..... so S=2 and the answer to the question is 3+8+1+2 = 14

Interesting that the resulting sum stayed exactly the SAME. There are only a few options left, but if you map them out, you'll end up with the exact same answer every time... the answer to the question is ALWAYS 14.
Fact 1 is SUFFICIENT

2) P, Q, R, and S are distinct non-zero digits.

With this Fact, we know that the 4 digits are all DIFFERENT non-zero integers, but there are multiple possible answers to the given question.
Fact 2 is INSUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

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