Problem Solving

Hello,

I am stuck with #38 data sufficiency of OG12.

Here's the question:

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?

Rephrase:

Ratio: p1 / r1 > or < p2 / r2

1. p1 > p2 (insufficient since no mention of r1 and r2)

2. r2 > r1 (insufficient since no mention of p1 and p2)

1 and 2 combined

I know that since r2 > r1 then (1/r2) < (1/r1) since we know that r2 and r1 are positive

Now I am stuck. What do I do now with p1 > p2?

Please help, experts.

Thank you!

Hi David,

p1 > p2
r2 > r1

1/r2 < 1/r1

Multiplying both sides by p1,
p1/r2 < p1/r1 -- (1)

Since p2 < p1,
p2/r2 < p1/r2 -- (2)

Combining (1) and (2), we have

p2/r2 < p1/r2 < p1/r1
or
p2/r2 < p1/r1

[spoiler](C)[/spoiler] is the answer

Is the answer C

A and B are not the answers for obvious reasons as you mentioned
Now, since p1>p2 and (1/r1)>(1/r2)
we can easily state p1/r1>p2/r2 as both the values are positive and non-zero.

ankita1709 wrote:Is the answer C

A and B are not the answers for obvious reasons as you mentioned
Now, since p1>p2 and (1/r1)>(1/r2)
we can easily state p1/r1>p2/r2 as both the values are positive and non-zero.

Yes, the answer is C.

But can you elaborate more how did you arrive from p1 > p2 and (1/r1) > (1/r2) to p1/r1 > p2/r2

I know I am lacking something.

Hi,

I just explained it above. Please let me know if it makes sense.

Lets make a small change in the problem.
With all the other information same, if we had r1 > r2 in Statement(2) what would the answer have been?

mdavidm_531 wrote:Hello,

I am stuck with #38 data sufficiency of OG12.

Here's the question:

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?

Rephrase:

Ratio: p1 / r1 > or < p2 / r2

1. p1 > p2 (insufficient since no mention of r1 and r2)

2. r2 > r1 (insufficient since no mention of p1 and p2)

1 and 2 combined

I know that since r2 > r1 then (1/r2) < (1/r1) since we know that r2 and r1 are positive

Now I am stuck. What do I do now with p1 > p2?

Please help, experts.

Thank you!

Hi!

When you're not sure what to do with the algebra, don't waste your valuable test time trying to figure it out. Instead, dive right in and pick numbers.

As you noted, each statement must be insufficient alone, since each one only provides 1/2 the picture. So, let's jump right to combination.

1) p1>p2
2) r2>r1

We have to pick numbers that follow the rules given, so let's try:

p1 = 4
p2 = 3
r2 = 2
r1 = 1

Now when we plug into the ratio, we get:

p1/r1 = 4/1 = 4
and
p2/r2 = 3/2 = 1.5

Is 4 > 1.5? YES

At this point we have two options: we can try to change the numbers to get a different answer or we can reason out whether we'll always get a YES answer.

Let's reason it out, now that we have some concrete info to help.

The numerator of p1/r1 is greater than the numerator of p2/r2; the denominator of p1/r1 is smaller than the denominator of p2/r2. Since increasing the numerator and decreasing the denominator are both ways to INCREASE the value of a fraction, p1/r1 will always be bigger than p2/r2. Sufficient, choose C!

If we were going to try to pick other numbers, we'd soon see that there's no permissible numbers we can pick to make p2/r2 the bigger ratio - after a couple of tries we'd give up and say that p1/r1 will always be bigger.

Plugging in numbers is good but in my opinion this is the easiest and the most convincing method of all:



Not the best job at drawing, but I hope you get the point.

Please feel free to check my explanation here:

https://www.beatthegmat.com/p1-and-p2-t108479.html

I think I just figured it out while on my way home. lol

Anyway,

Rephrase: p1/r1 > p2/r2 or p1/r1 < p2/r2

1. p1 > p2 - insufficient (obvious)
2. r2 > r1 - insufficient (obvious)

1. and 2. together

Now here comes the intuitive thinking part.

I look back at fractions and realize that for any fraction, say p1/r1:

The value of p1/r1 increases when ANY of the following happens: (1) p1 increases OR (2) r1 decreases OR both (1) and (2) happen.

Now, it says in statement 1 that p1 > p2. This means that the NUMERATOR of p1/r1 is greater than the NUMERATOR of p2/r2.

But we shouldn't stop here since we don't know anything about the denominator. Hence, we move to statement 2.

As such, statement 2 says that r2 > r1 or r1 < r2. This means that the DENOMINATOR of p1/r1 is less than the DENOMINATOR of p2/r2.

So the combo situation happens here: the numerator becomes bigger while the denominator becomes smaller. As such, p1/r1 is indeed greater than p2/r2.

I just noticed that this thinking is also applicable to critical reasoning, specifically arguments that use numbers. For example, an increase in production per worker (mathematically: total production / number of workers) could be interpreted as: (1) what if the number of workers decreased but same total production or (2) what if the number of workers is the same but the total production increased.

See, there's a bridge that connects quant and verbal. Or am I just pushing things? I hope not.

aneesh.kg wrote:Plugging in numbers is good but in my opinion this is the easiest and the most convincing method of all:



Not the best job at drawing, but I hope you get the point.

This is great aneesh.kg