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If \(p_1\) and \(p_2\) are the populations and \(r_1\) and \(r_2\) are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?

(1) \(p_1>p_2\) (2) \(r_2>r_1\)

Practice Questions Question: 43 Page: 278 Difficulty: 600

If \(p_1\) and \(p_2\) are the populations and \(r_1\) and \(r_2\) are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?

Question asks whether \(\frac{p_1}{r_1}>\frac{p_2}{r_2}\). Or, since the numbers are positive, we are asked to determine whether \(p_1*r_2>p_2*r_1\).

(1) \(p_1>p_2\). No info about \(r_1\) and \(r_2\). Not sufficient. (2) \(r_2>r_1\). No info about \(p_1\) and \(p_2\). Not sufficient.

(1)+(2) Since the multiples on left hand side are greater than the respective multiples on the right hand side then \(p_1*r_2>p_2*r_1\). Sufficient.

If \(p_1\) and \(p_2\) are the populations and \(r_1\) and \(r_2\) are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?

(1) \(p_1>p_2\) (2) \(r_2>r_1\)

Practice Questions Question: 43 Page: 278 Difficulty: 600

Each week we'll be posting several questions from The Official Guide for GMAT® Review, 13th Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

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The question asks to compare to fractions: \(\frac{p_1}{r_1}\) and \(\frac{p_2}{r_2}\), where all 4 numbers are positive integers.

(1) Not sufficient, because we don't have any information about the denominators of the two fractions to be compared. For example, we can choose \(p_1=10r_1 and p_2=100r_2\) or the other way around. (2) Again, not sufficient, because now we don't have any information on the numerators. We can choose again the same values for \(p_1\) and \(p_2\) as above.

(1) and (2) together: We know that the numerator of the first fraction \(\frac{p_1}{r_1}\) is greater than the numerator of the second fraction \(\frac{p_2}{r_2}\). In addition, the denominator of the first fraction is smaller than the denominator of the second fraction. Therefore, the first fraction is greater than the second, because \(\frac{p_1}{r_1}>\frac{p_2}{r_1}>\frac{p_2}{r_2}\). Between two positive fractions with the same numerator, the largest fraction is that with the smallest denominator.

Answer C.
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Re: If p1 and p2 are the populations and r1 and r2 are the [#permalink]

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18 Sep 2012, 05:56

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If p_1 and p_2 are the populations and r_1 and r_2 are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts? (1) p_1>p_2 (2) r_2>r_1

The Question can be restated as which one is greater \(p_1.r_2 or p_2.r_1\) 1) No info is given regarding the ratio of \(r_2 & r_1\) ---->Insufficient 2) No info is given regarding the ratio of \(p_2 & p_1\) ---->Insufficient 1+2) We can easily say that \(p_1.r_2\) is greater than \(p_2.r_1\)-->Sufficient

Answer C

Note:- If the option (2) had been \(r_2<r_1\) rather than \(r_2>r_1\), then the answer would have been E

Hope it helps.
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If \(p_1\) and \(p_2\) are the populations and \(r_1\) and \(r_2\) are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?

(1) \(p_1>p_2\) (2) \(r_2>r_1\)

Practice Questions Question: 43 Page: 278 Difficulty: 600

Each week we'll be posting several questions from The Official Guide for GMAT® Review, 13th Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project: 1. Please provide your solutions to the questions; 2. Please vote for the best solutions by pressing Kudos button; 3. Please vote for the questions themselves by pressing Kudos button; 4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!

Need to find is P1/R1 > P2/R2

Option 1: P1>p2 only by judging neumerator we can not conclude which ratio is greator the other. therefore Option 1 is not sufficient to answer the question. Option 2: R2>R1 again the same thing only by looking at the denominator we can not say that which ratio is greator. therefor Option 2 is also not sufficient to answer the question.

by combining both the option it is coming that Neumerator for the P1/R1 is greator than P2/R2 and Denominator of P1/R1 is lesser than the Denominator of P2/R2. therefor P1/R1 is greator the P2/R2.

Re: If p1 and p2 are the populations and r1 and r2 are the [#permalink]

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20 Sep 2012, 00:24

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Is P1/R1 > P2/R2, or vise versa? let's manipulate the inequality to a simpler form. P1R2 > P2R1 means P1/R1 is the greater ratio?

(1) P1 > P2, we can't answer the question because we need to know R1,R2. INSUFFICIENT (2) r2 > R1, we can't answer the question because we need to know P1, P2. INSUFFICIENT.

Together, let's multiple the inequalities. p1r2 > p2r1', so we now know that P1/r1 is the greater ratio.

If \(p_1\) and \(p_2\) are the populations and \(r_1\) and \(r_2\) are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?

Question asks whether \(\frac{p_1}{r_1}>\frac{p_2}{r_2}\). Or, since the numbers are positive, we are asked to determine whether \(p_1*r_2>p_2*r_1\).

(1) \(p_1>p_2\). No info about \(r_1\) and \(r_2\). Not sufficient. (2) \(r_2>r_1\). No info about \(p_1\) and \(p_2\). Not sufficient.

(1)+(2) Since the multiples on left hand side are greater than the respective multiples on the right hand side then \(p_1*r_2>p_2*r_1\). Sufficient.

Answer: C.

Kudos points given to everyone with correct solution. Let me know if I missed someone.
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If p1 and p2 are the populations and r1 and r2 are the [#permalink]

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29 Dec 2012, 12:40

I took this approach 1. reads P1>P2 and rewrote 2. as R1<R2

Each statement by itself is not sufficient. Combining- then P1/R1 (greater numerator/lower denominator ) > P2/R2 (lower numerator/greater denominator) will be true.

If p1 and p2 are the populations and r1 and r2 are the [#permalink]

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31 Dec 2012, 12:10

metallicafan wrote:

Which way is better to solve this question? Using algebra or testing with some values or numbers? Why? I used numbers, but OG used algebra.

If P1 and P2 are the populations and R1 and R2 are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts? (1) P1 > P2 (2) R2 > R1

Testing with Numbers if one can quickly come up with a yes and another no is always helpful on GMAT. But key is to find a contrast. Also it will help if a pattern emerges in a few steps. Usually GMAT doesn't something that holds for x upto say 20 terms and fails on 23rd.

If P1 and P2 are the populations and R1 and R2 are the numbers o [#permalink]

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07 Feb 2013, 10:57

ksrajgmat wrote:

Can some one please explain the solution to this problem.

I used substitution for this problem.

St 1 gives us that P1 > P2, for ease of math I subbed P1 = 100 P2 = 50 --> This is NS to solve the problem since we don't know how many Reps there are for each district & therefore cannot find the ratio. --> Eliminate A & D

St 2 gives us that R2 > R1, for ease of math I subbed R1 = 10 R2 = 20 --> This is NS also, since we don't know the population --> Eliminate B

Combining the 2 statements we see that P1 > P2 and R2 > R1 -- at this point, even without substitution you should intuitively know that a lower population with a higher pop count will give you the biggest ratio (aka the higher percentage), but just to make sure I will use the same numbers I substituted above.

So, R1/P1 = 10/100 = 1/10 (or 10%...whichever form is easiest for you to work) and R2/P2 = 20/50 = 2/5 (or 40%)...Sufficient....so C

As a reminder, with DS you don't actually have to solve the problem - you just need to be able to recognize whether you have enough information to solve it.

Re: If p1 and p2 are the populations and r1 and r2 are the [#permalink]

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If \(p_1\) and \(p_2\) are the populations and \(r_1\) and \(r_2\) are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?

(1) \(p_1>p_2\) (2) \(r_2>r_1\)

Solution:

We are given the following:

p_1 = population of District 1

p_2 = population of District 2

r_1 = the numbers of representatives of District 1

r_2 = numbers of representatives of District 2

We need to determine whether the ratio of the population to the number of representatives is greater in District 1 or District 2. We can translate the question into an inequality.

Is p_1/r_1 > p_2/r_2 ?

After cross multiplying we obtain:

Is (p_1)(r_2) > (r_1)(p_2) ?

Note that we could write the initial equation as p_1/r_1 < p_2/r_2 as well, because the question is only asking which one is greater. Whichever way we write the equation would be acceptable.

Statement One Alone:

p_1 > p_2

Although p_1 > p_2, we do not have enough information to determine whether (p_1)(r_2) is greater than (r_1)(p_2). Let’s consider two cases.

Case # 1

p_1 = 300

p_2 = 200

r_1 = 2

r_2 = 1

We see that (p_1)(r_2) > (r_1)(p_2) = 300 > 400 is not true.

Case # 2

p_1 = 300

p_2 = 200

r_1 = 2

r_2 = 2

We see that (p_1)(r_2) > (r_1)(p_2) = 600 > 400 is true.

Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

r_2 > r_1

Although r_2 > r_1, we do not have enough to determine whether (p_1)(r_2) is greater than (r_1)(p_2). Let’s consider two cases.

Case # 1

p_1 = 100

p_2 = 200

r_1 = 2

r_2 = 3

We see that (p_1)(r_2) > (r_1)(p_2) = 300 > 400 is not true.

Case # 2

p_1 = 200

p_2 = 200

r_1 = 2

r_2 = 3

We see that (p_1)(r_2) > (r_1)(p_2) = 600 > 400 is true.

Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information from statements one and two we know the following:

p_1 > p_2 and r_2 > r_1. Thus, (p_1)(r_2) must be greater than (r_1)(p_2).

The answer is C.
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GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions

Re: If p1 and p2 are the populations and r1 and r2 are the [#permalink]

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09 Aug 2016, 11:50

Bunuel wrote:

If \(p_1\) and \(p_2\) are the populations and \(r_1\) and \(r_2\) are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?

(1) \(p_1>p_2\) (2) \(r_2>r_1\)

Practice Questions Question: 43 Page: 278 Difficulty: 600

We are interested in a ratio of either P1/R1 or P2/R2. The statements provide P1 > P2 and R1 > R2.

What if P1 = 6 and P2 = 3? P1 > P2. What if R1 = 12 and R2 = 6? R1 > R2.

But it still doesn't mean P1/R1 > P2/R2. They can also be the equal. So to me the answer should be (E).

If \(p_1\) and \(p_2\) are the populations and \(r_1\) and \(r_2\) are the numbers of representatives of District 1 and District 2, respectively, the ratio of the population to the number of representatives is greater for which of the two districts?

(1) \(p_1>p_2\) (2) \(r_2>r_1\)

Practice Questions Question: 43 Page: 278 Difficulty: 600

We are interested in a ratio of either P1/R1 or P2/R2. The statements provide P1 > P2 and R1 > R2.

What if P1 = 6 and P2 = 3? P1 > P2. What if R1 = 12 and R2 = 6? R1 > R2.

But it still doesn't mean P1/R1 > P2/R2. They can also be the equal. So to me the answer should be (E).

Where am I wrong in my assumption?

(2) says \(r_2>r_1\) not \(r_1>r_2\)
_________________

Re: If p1 and p2 are the populations and r1 and r2 are the [#permalink]

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10 Aug 2016, 03:11

The question asks if P1/R1 > P2/R2 or opposite. (1) p1>p2 => INSUFFICIENT (2) r2>r1 => INSUFFICIENT

(1) and (2): (1) p1 > p2 (2)1/r1>1/r2 (because r2>r1) Multiply (1) and (2) as above, we have p1/r1 > p2/r2 => SUFFICIENT => C is the right answer

However, if the (2) is r1>r2 instead of r2>r1, we will have E as the answer. Because in that case we have p1/r2>p2/r1 and we have no way to identify if p1/r1 is larger than p2/r2 or not.

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