A country's per capita national debt is its national debt divided by

Let D be the debt and let P be the population. The original question: Is \(495\leq \frac{D}{P}\leq 505\) ?

1) We know that \(42{,}500{,}000{,}000\leq D<43{,}500{,}000{,}000\), but no information is given about the population. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

2) We know that \(85{,}500{,}000\leq P<86{,}500{,}000\), but no information is given about the debt. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

1&2) First, we check whether the maximum possible value for D/P meets the upper bound condition in the original question.

Is \(\frac{435{,}000}{855}\leq 505\) ? To answer the question, we should use multiplication rather than division.

Since (855)(505)=431,775 is not greater than 435,000 , the maximum possible value for D/P is greater than the upper bound in the original question.

Then, we check whether the minimum possible value for D/P meets the upper bound condition in the original question. It's important that we check the upper bound condition, not the lower bound condition. Imagine that it could meet the lower bound condition with or without meeting the upper bound condition.

Is \(\frac{425{,}000}{865}\leq 505\) ?

Since (865)(505)=436,825 is greater than 425,000 , the minimum possible value for D/P is not greater than the upper bound in the original question.

Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

Answer: E

To make the math easier, let's ignore the last six 0's in each statement, as follows:
1. Country G's national debt to the nearest $1,000 is $43,000
2. Country G's population to the nearest integer is 86

Each statement on its own is clearly insufficient.

Statements combined:
Case 1: National debt = 43,000 and population = 86
Per capita debt \(= \frac{43,000}{86} = 500\)
In this case, the answer to the question stem is YES.

The question stem will be NO if the per capita debt exceeds 505.
Since \(\frac{43,000}{86} = 500\), \(\frac{43,000}{86} + \frac{5.1*86}{86} = 500 + 5.1 = 505.1\)

Case 2: National debt = 43,000 + 5.1*86 = 43,438.6 and population = 86
As shown in the blue statement above, per capita debt = 505.1
In this case, the answer to the question stem is NO.

Since the answer is YES In Case 1 but NO in Case 2, the two statements combined are INSUFFICIENT.


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ZoltanBP
Hello
May I know Why did you use ≤ D < ?
Why not it is both ≤ or < at a time?
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Hello Bunuel
May i have your best efforts here in this question? Your help will be appreciated..
Thanks__
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Hi Asad,

I used different signs because of the rounding rule.

42,500,000,000 rounded to the nearest 1,000,000,000 is 43,000,000,000. However, 42,499,999,999 rounded to the nearest 1,000,000,000 would only be 42,000,000,000. That's why I used the \(\leq\) sign for the lower bound.

43,500,000,000 rounded to the nearest 1,000,000,000 would be an excessive amount of 44,000,000,000. However, 43,499,999,999 rounded to the nearest 1,000,000,000 is 43,000,000,000. That's why I used the < sign for the upper bound.

Using these signs, I could keep the nice numbers in the inequality.

The reason for not using less than equal to sign in the upper bound is that the upper bound integer would round off to the higher billion/million. For eg: 43,500,000,000 would round off to 44,000,000,000

Hi Zoltan,

Why do you check whether the lowest possible value for D/P meets the upper bound condition instead of the lower bound? I agree with your statement "the minimum possible value for D/P is not greater than the upper bound in the original question" but, I don't see how this means we can't get a definite answer to the original question. I'm not saying your answer is incorrect, I own a GMAT Official Advanced Questions and I know the answer is most definitely E. I was looking for another method to answer the question different from the one in the book because I think it entails too many unnecessary calculations and I thought yours was a neat and efficient approach but I don't understand that specific part. In the GMAT Off AQ answer explanation it actually proves that the lowest possible value is under the lower bound and that the highest possible value is over the upper bound.

Thanks in advance.

oe

The only reason we can't have definite answer is because
The Answer ranges from around 491<X<508.
We only need answer range till 495≤x≤505
Whereas the answer can take more than 1 value,
On ending note, We don't have to check whether the answer is possible or not. We have check whether only the said answer is possible or not.

Posted from my mobile device
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This cannot be done in 2 minutes. What is the best approach to do this type of question if encounter in GMAT.please suggest

Posted from my mobile device

chetan2u ,
VeritasKarishma

hey there

I picked E, just want to make sure my reasoning is correct

I didnt quite understood expression "within $5 of $500" though i interpreted is as 5/500 which is 0.01 so per capita

ST 1. (1) Country G's national debt to the nearest billion

I didn`t make any calculations here, just my thought process was that if number is rounded to the nearest 1 billion than number could vary substantially whereas 5/500 is tiny number compared to billion and when rounding to billion the range of national debt compared that of rounded one can vary substantially

So same reasoning is applied to the second statement

Is my reasoning correct ? perhaps partially

yashikaaggarwal whats wrong with my post letters look strange can you pls format it so it look decent

Hi

Can you explain how did you get that 5.1? And why do we multiply it with 86? I know we are trying to decode the range but can you explain it clearer. Thank you.

Question: Is the per capita debt within $5 of $500?

Case 1:
When the national debt = 43,000 and the population = 86, the per capita debt = \(\frac{43,000}{86}\) = 500, so the answer to the question stem is YES.

Since Case 1 yields an answer of YES, the goal in Case 2 is to get an answer of NO.
The answer to the question stem will be NO if the per capita debt exceeds 505.
Thus, the goal in Case 2 is to yield the following per capita debt:
500 + 5.1

To yield the value in green, I chose to change only the numerator in the blue fraction above, while retaining the denominator of 86.
\(\frac{86}{86} = 1\)
Thus, \(5.1 = 5.1 * \frac{86}{86} = \frac{5.1*86}{86}\)

Since \(500 = \frac{43,000}{86}\) and \(5.1 = \frac{5.1*86}{86}\), we get:
\(505.1 = \frac{43,000}{86} +\frac{ 5.1*86}{86} = \frac{43,438.6}{86}\)

The result in Case 2 is a numerator -- and thus a national debt -- that satisfies the rounding condition in Statement 1.
Case 2: National debt = 43,438.6, population = 86, per capita debt \(= \frac{43,438.6}{86} = 505.1\)
In this case, the per capita debt exceeds $505, so the answer to the question stem is NO.
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Hello GMATGuruNY,

I was wondering whether you could help me on this one,
I have seen posts that examine various cases to answer the question and calculate those cases so I was wondering whether my approach has a flaw that I'm not aware of.

What I did, was to divide 43.000/86=500 we want to see whether the value is 5<x<500 , so basically if we add even a single unit to 43.000 (43.001) then x>500 so the answer is NO, but if we add even a single value to 500(501) then x<500 so the answer is YES , thus the answer is E , so is the above approach ok or is there something that I'm missing?

The expression in red -- which represents a value between 5 and 500 -- leads me to believe that you have misinterpreted the question stem.
Question stem:
Is the per capita national debt of Country G within $5 of $500?
WITHIN $5 OF $500 does not mean BETWEEN 5 AND 500.
within $5 of $500 means that the difference between the per capita debt and 500 is less than or equal to 5.
In terms of math:
495 ≤ per capita debt ≤ 505
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Let's see...

43000/86 = 500, a good start.

Now if we minimize the numerator and maximize the denominator, we get infinitely close to 42500/86,5.

86,5 * 500 = 43 250
86,5 * 495 = 43 250 - 86,5*5 = ~42 820

This should imply that the extreme case of 42 500/86,5 would give us a per capita debt of less than 495. We can conclude that the per capita debt may or may not fall within 5$ of 500$.

Answer: E

This is a high quality question by GMAC

A country's per capita national debt is its national debt divided by its population.

Clearly each statement alone is not sufficient.

Combined:

We can ignore the last six zero's in the national debt and the population.

\(\frac{43,000 }{ 86}\) = 500

Now, we can either adjust the national debt or the population. However, it's easier to adjust the national debt in this situation and leave the denominator as 86.

The national debt is between 42,500 and 43,500.

43,500 is 500 more than 43,000. We can simply divide 500 by 86:

\(\frac{500}{86}\) > 5

We have a yes and no case -- the answer is E.
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