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

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Math Expert
Joined: 02 Sep 2009
Posts: 44636
PQ and QP represent two-digit numbers having P and Q as their digits. [#permalink]

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28 Jul 2017, 01:21
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Difficulty:

95% (hard)

Question Stats:

48% (02:01) correct 52% (02:27) wrong based on 158 sessions

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PQ and QP represent two-digit numbers having P and Q as their digits. RSR is a three-digit number having the digits R and S. What is the value of P + Q + R + S?

(1) PQ + QP = RSR.
(2) P, Q, R and S are distinct non-zero digits.
[Reveal] Spoiler: OA

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Joined: 18 Aug 2016
Posts: 631
Re: PQ and QP represent two-digit numbers having P and Q as their digits. [#permalink]

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28 Jul 2017, 01:38
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Bunuel wrote:
PQ and QP represent two-digit numbers having P and Q as their digits. RSR is a three-digit number having the digits R and S. What is the value of P + Q + R + S?

(1) PQ + QP = RSR.
(2) P, Q, R and S are distinct non-zero digits.

(1)

R cannot be 2 and has to be 1 as sum of 2 two digit numbers cannot exceed 198 (99+99)

SO now since R is 1 S can take 1, 2, 3, 4, 5, 6, 7, 8, 9, 0

but since PQ and QP = RSR

P and Q cannot take 0
and one can only be odd digit and one even
Also one of them has to be greater than 5

92 + 29 = 121
83 + 38 = 121
74 + 47 = 121

in all cases P+Q = 11 an R+S = 3
Sufficient

(2) Not sufficient ( We cannot take same values of P and Q)

A
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Re: PQ and QP represent two-digit numbers having P and Q as their digits. [#permalink]

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31 Aug 2017, 04:44
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PQ and QP can be written in following ways

PQ = 10P+Q (Example: 21=10(2)+1)
QP = 10Q+P

PQ+QP = 10P+Q+10Q+P = 11(P+Q)
Given that PQ+QP = RSR

so, RSR = 11(P+Q)
-> RSR is a multiple of 11

Since it is given that the unit's digit and the hundred's digit are same i.e R
The need to be multiplied by either 11 (11*11 = 121), 22(11*22 = 242), 33....
But P+Q cannot exceed 18(as the max possible single digit integer is 9)

So, P+Q = 11 and RSR = 121
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Re: PQ and QP represent two-digit numbers having P and Q as their digits. [#permalink]

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28 Oct 2017, 22:38
Bunuel wrote:
PQ and QP represent two-digit numbers having P and Q as their digits. RSR is a three-digit number having the digits R and S. What is the value of P + Q + R + S?

(1) PQ + QP = RSR.
(2) P, Q, R and S are distinct non-zero 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
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Re: PQ and QP represent two-digit numbers having P and Q as their digits. [#permalink]

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16 Dec 2017, 15:46
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

[Reveal] Spoiler:
A

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Re: PQ and QP represent two-digit numbers having P and Q as their digits.   [#permalink] 16 Dec 2017, 15:46
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