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Intern  S
Joined: 10 Jul 2016
Posts: 44
If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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15 00:00

Difficulty:   95% (hard)

Question Stats: 52% (02:39) correct 48% (02:33) wrong based on 157 sessions

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If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and if P>Q, which of the following could be the value of R?

A. 50
B. 65
C. 75
D. 90
E. 100
Math Expert V
Joined: 02 Aug 2009
Posts: 7984
If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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siddharthsinha123 wrote:
If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and if P>Q, which of the following could be the value of R?

A. 50
B. 65
C. 75
D. 90
E. 100

Hi,
Easiest way would be to substitute choices in given equation 50p+100q=r(p+q) and check for P>Q..

A. 50.....$$50p+100q=50(p+q)....50p+100q=50p+50q....100q=50q$$... Wrong
B. 65.....$$50p+100q=65(p+q)....50p+100q=65p+65q....35q=15p... p=\frac{35q}{15}...P>Q$$.. Correct
C. 75.....$$50p+100q=75(p+q)....50p+100q=75p+75q....25q=25p.. P=Q$$.... Wrong
D. 90.....$$50p+100q=90(p+q)....50p+100q=90p+90q....10q=40p.. ..P<Q$$..Wrong
E. 100.....$$50p+100q=100(p+q)....50p+100q=100p+100q....100p=50p....P=0$$... Wrong

Only B is correct
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Joined: 17 Dec 2016
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If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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Hi

Simplest method would be to re-arrange the given statement as :

(50P + 100Q)/(P+Q) = R

From this you know that this is nothing but weighted average. It is given that P > Q, hence the average would be closer to 50 than to 100 and not in the middle value (75). You are left with only 65 (B).

Thanks
Rohit
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Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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siddharthsinha123 wrote:
If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and if P>Q, which of the following could be the value of R?

A. 50
B. 65
C. 75
D. 90
E. 100

I got to B easily through few steps...
100Q-RQ = RP - 50P
Q(100-R) = P(R-50)
from the above, we can get:
50<R<100
A and E are right away out.

if R =90, then 100-R=10 and 100-50 = 50, but this doesn't satisfy the condition P>Q, but rather vice versa Q>P - D is out.

at least we got now a 50% chance to get the right answer, m? :D

let's try C:
100-75=25
75-50=25

25Q=25P -> P=Q -> doesn't satisfy the condition P>Q

the only option left is B.
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Joined: 15 Jun 2013
Posts: 46
Schools: Ivey '19 (I)
GMAT 1: 690 Q49 V35 GPA: 3.82
Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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siddharthsinha123 wrote:
If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and if P>Q, which of the following could be the value of R?

A. 50
B. 65
C. 75
D. 90
E. 100

From the equation above we see that P(50-R)+Q(100-R)=0.
Therefore R<>50 or 100, so A and E are out. Also, P<>Q so C is out. Now we see that P will be with negative sign and Q - with Positive. Taking into account that P>Q |50-R| should be <|100-R|. Thus B is the answer |50-65|=15<|100-65|=35 (with D it would be |50-90|=40>|100-90|=10.
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Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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1
Attachment: FullSizeRender (4).jpg [ 64.61 KiB | Viewed 4596 times ]

ANS: B
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Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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Leo8 wrote:
Attachment:
FullSizeRender (4).jpg

ANS: B

One doubt...

P(50-R) = Q(R-100)
P/Q=(R-100)/(50-R)------(1)
Also, P>Q
IMPLIES P/Q>1
BUT FROM eq (1), this means
(R-100)/(50-R) > 1
R-100>50-R
2R>150
R>75

Which is opposite of what you have shown.

Can you spot the issue with my approach??
Manager  B
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Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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On substituting the equation R=65 then the we have 35q=15p... or p=35q/15...P>Q..
But this value can be non integer if q=1,2.... etc then doesn't it violate the statement that P,Q and R are positive integers.

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Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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$$50P + 100Q = R(P + Q)$$

$$50(P + 2Q) = R(P + Q)$$

$$50[(P + Q) + Q] = R(P + Q)$$

$$50Q = (R - 50)(P + Q)$$

$$\frac{Q}{(P + Q)}= \frac{(R - 50)}{50}$$

Let's try with the answer in the middle (C) $$R=75 => \frac{(75 - 50)}{50}= \frac{25}{50} = \frac{1}{2} => \frac{Q}{(P + Q)}= \frac{1}{2}$$ => Q=1 and P=1, which cannot be (P must be greater than Q). With this, the bigger answers D and E (90 and 100) cannot be either.

Let's with B: $$R=65 => \frac{(65 - 50)}{50}= \frac{15}{50} = \frac{3}{10} => \frac{Q}{(P + Q)}= \frac{3}{10}$$ => Q=3 and P=7, and this matches with the statement which says that P > Q. => Correct Answer: B.
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If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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Can be easily solved using weight average concept.

Given: 50 ----------------R------------------100
weight:P--------------------------------------Q
if P = Q, then R will be at centre i.e = 75
however given, P > Q, weighted average will be pulled towards 50 and will be less than 75
only choice B fits perfectly, as weighted average can't be equal to 50 (since P = 50)

Kudos please, if you like my approach, need to unlock my gmat club tests

Thanks
Manager  B
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Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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Leo8 wrote:
Attachment:
FullSizeRender (4).jpg

ANS: B

can u explain how you got 50-r as greater
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Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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rahulkashyap wrote:
Leo8 wrote:
Attachment:
FullSizeRender (4).jpg

ANS: B

can u explain how you got 50-r as greater

from our equation :

P(50 -R) = Q( R - 100)

But Its given P > Q so for the equality to be true the co-effficent of P must be greater than co-efficient of Q

50 - R > R - 100

also nitesh181989 :

P/q > 1 will give you equation : 50 - R > R - 100

you have put co-effiecient of Q in the numerator and that of P in denominator
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Manager  B
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Posts: 225
Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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Leo8 wrote:
rahulkashyap wrote:
Leo8 wrote:
Attachment:
FullSizeRender (4).jpg

ANS: B

can u explain how you got 50-r as greater

from our equation :

P(50 -R) = Q( R - 100)

But Its given P > Q so for the equality to be true the co-effficent of P must be greater than co-efficient of Q

50 - R > R - 100

also nitesh181989 :

P/q > 1 will give you equation : 50 - R > R - 100

you have put co-effiecient of Q in the numerator and that of P in denominator

Well, first of all if p if greater than q, the inequality sign should be there other way. Not what you've pointed. (bigger) x (smaller)= (smaller) x(bigger)
Even then, you don't know whether lhs and rhs are both positive / negative. So you can't say with certainty

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Re: If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and  [#permalink]

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