If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and

Question

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

rohitchauhan10 · Accepted Answer

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

Leo8 · Answer

FullSizeRender (4).jpg ANS: B

chetan2u · Answer

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

mvictor · Answer

I got to B easily through few steps... 100Q-RQ = RP - 50P Q(100-R) = P(R-50) from the above, we can get: 50Q, 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.

malevg · Answer

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.

nitesh181989 · Answer

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??

FB2017 · Answer

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.

Please advise.

varodriguezh · Answer

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

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

50  = 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.

hellosanthosh2k2 · Answer

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

rahulkashyap · Answer

can u explain how you got 50-r as greater

Leo8 · Answer

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

rahulkashyap · Answer

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

Posted from my mobile device

avirupm97 · Answer

Hi  chetan2u , Might be silly but would this approach work as well?

=50p+100q=r(p+q)
=50p + 100q/(p+q) = r
=50p + 50q + 50q/(p+q) = r
=50 + 50q/(p+q) = r ......(The answer will be > 50 but always less than 75 because 50q/(p+q) can never be >=25 since p>q )

chetan2u · Answer

That’s a perfectly fine solution. No error at all.

Kinshook · Answer

Asked: 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?

50P + 100Q = R (P + Q)
(50-R)P + (100-R)Q = 0
P = (100-R)Q/(R-50)
Since P > Q
(100-R)Q/(R-50) > Q
(100-R) > R-50
R < 75

Since P > 0
(100-R)Q/(R-50) > 0
R < 100
R > 50

IMO B

Sunpreet19 · Answer

Posted from my mobile device

The solution you’ve posted, you went wrong in third last step. (R-100)/(50-R) >1 we cannot cross multiply since we’re unaware of the sign of (50-R) which will depend on value of R . We shift 1 to LHS ie 
R-100 
———- -1 > 0
50-R

2R-150
—————>0
50-R

Now substitute values of R to check

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If P,Q and R are positive integers such that 50P + 100Q = R(P + Q) and

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