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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: 42264

Kudos [?]: 132771 [1], given: 12372

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% (02:14) correct 52% (02:24) wrong based on 115 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

_________________

Kudos [?]: 132771 [1], given: 12372

Director
Joined: 18 Aug 2016
Posts: 550

Kudos [?]: 161 [3], given: 134

GMAT 1: 630 Q47 V29
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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Luckisnoexcuse

Kudos [?]: 161 [3], given: 134

Manager
Joined: 27 Jan 2016
Posts: 150

Kudos [?]: 67 [0], given: 124

Schools: ISB '18
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

Kudos [?]: 67 [0], given: 124

Director
Joined: 25 Feb 2013
Posts: 560

Kudos [?]: 256 [0], given: 34

Location: India
Schools: Mannheim"19 (S)
GPA: 3.82
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

Kudos [?]: 256 [0], given: 34

Re: PQ and QP represent two-digit numbers having P and Q as their digits.   [#permalink] 28 Oct 2017, 22:38
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