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Math Expert V
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If p1 and p2 are the populations and r1 and r2 are the numbers of repr  [#permalink]

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Question Stats: 74% (01:23) correct 26% (01:23) wrong based on 872 sessions

### HideShow timer Statistics 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

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

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SOLUTION

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.

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

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Bunuel wrote:
The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

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

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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.

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Re: If p1 and p2 are the populations and r1 and r2 are the numbers of repr  [#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

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

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

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Bunuel wrote:
The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

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

GMAT Club is introducing a new project: The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

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:
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.

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

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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.

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

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Simple logic can help:

ratio is a fraction and for the less denominator and more numerator we get higher fraction

St.1 gives P1>P2, numerator is higher in the first pair
St.2 gives R2>R1, denominator is less in the first pair

so the first pair has higher ratio, C
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Re: If p1 and p2 are the populations and r1 and r2 are the numbers of repr  [#permalink]

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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) $$r_2>r_1$$

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).

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GMAT 1: 750 Q49 V43 Re: If p1 and p2 are the populations and r1 and r2 are the numbers of repr  [#permalink]

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MA 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) $$r_2 > r_1$$

Official Guide 12 Question Question: 38 Page: 276 Difficulty: 600

Find All Official Guide Questions

Video Explanations:

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) $$p_1 > p_2$$
Only population info , no info about representatives
Clearly INSUFFICIENT

(2) $$r_2 > r_1$$
Only representatives, no info about population
Clearly INSUFFICIENT

MERGE both statements
Ratio is just a fancy division represented as a fraction, rather than quotient and reminder form
By merging these 2 statements, we know a bigger number is being divided by smaller number and a smaller number is being divided by a bigger number,
Obviously the ratio will be higher for 1st where numerator is bigger and denominator is smaller . That is the first town with P1 and R1

HENCE SUFFICIENT

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

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Can some one explain me why the answer isn’t E

Let me assume some random values. Say p1=4 and p2=3. Since we have p1>p2 (4>3)
And r1>r2 say r1=8 and r2=6 (8>6)

Then we have the ratio 4/8>3/6 = 1/2>1/2 ??

If I take the ratio 4/8 and 3/7 then 0.5>0.42 ??

So we are not able to determine the exact answer !!

Posted from my mobile device
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Re: If p1 and p2 are the populations and r1 and r2 are the numbers of repr  [#permalink]

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MeghnaIjjapureddy wrote:
Can some one explain me why the answer isn’t E

Let me assume some random values. Say p1=4 and p2=3. Since we have p1>p2 (4>3)
And r1>r2say r1=8 and r2=6 (8>6)

Then we have the ratio 4/8>3/6 = 1/2>1/2 ??

If I take the ratio 4/8 and 3/7 then 0.5>0.42 ??

So we are not able to determine the exact answer !!

Posted from my mobile device

(2) says $$r_2>r_1$$ not vise-versa.
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Re: If p1 and p2 are the populations and r1 and r2 are the numbers of repr  [#permalink]

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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

One thing is for sure that we can't determine the ratio without numerator/denominator, we'll need both.

Now 1+2

Let's assume District 1 has higher ratio, let's see if we can validate it.

p1/r1 > p2/r2
Also, p1*r2 > p2*r1, this is possible because the factors on the left (p1,r1) are more than that of the right(p2,r2).
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I never gave up what I wanted- Re: If p1 and p2 are the populations and r1 and r2 are the numbers of repr   [#permalink] 21 May 2019, 21:41
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