If p1 and p2 are the populations and r1 and r2 are the numbers of repr

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.

Answer: C.
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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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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.

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.

Option 'C" is Answer.

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.

answer: C

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

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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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

ANSWER IS C

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

(2) says \(r_2>r_1\) not vise-versa.
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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).

Hello Bunuel IanStewart
Here, the question ask: Is \(\frac{p1}{r1}\)> \(\frac{p2}{r2}\)?
The highlighted part confused me . Need a bit clarification on it. It seems that IF r2>r1 (according to official statement), then the answer of that question YES. But, IF r1>r2 (according to highlighted part), the answer of that question is just NO (not YES and NO simultaneously)! So, it is still C. Am i missing anything?
Thanks__
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It's not a yes/no question, but the answer would indeed be E if Statement 2 was reversed. We'd then have this question:

If a, b, x and y are positive, which is greater, a/x or b/y ?
1. a > b
2. x > y

We might have a = 6, b = 3, x = 2 and y = 1. Then a/x and b/y are exactly equal (they are both 3). Now by making a slightly larger or slightly smaller, you can make a/x greater or less than b/y. So we don't have enough information to decide whether a/x is greater than b/y.

The answer to the original question is C though, since we're comparing a/x and b/y, and in the original version of the question, we know a > b and x < y, so a/x has both a larger numerator and smaller denominator than b/y, and so clearly must have a larger overall value.

\(\frac{P1}{P2}>\frac{P2}{R2}\)?

Cross-multiply...

\(P1 x R2 > P2 x R1\)?

(1) P1 > P2. One variable is greater than another. Not sufficient for four variables.
(2) R2 > R1. One variable is greater than another. Not sufficient for four variables.
(1+2) Both variables one one side of the equation are greater than both variables on the other side of the equation. Since all variables are positive, \(\frac{P1}{P2}>\frac{P2}{R2}\).

Clearly both statements alone are insufficient.

1+2 together:

\(p1/r1=p2/r2\)

just pick numbers as per the two given options: \(p1 > p2\) --> \(10 > 5\)

\(r2> r1\) --> \(12 > 10\)

\(P1= 10, p2=5, r1= 10, r2= 12\)

\(10/10 > 5 /12\) ?
\(1>0.4something\) ? YES !

Sufficient. C

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