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The functions \(f\) and \(g\) are defined for all the positive integers \(n\) by the following rule: \(f(n)\) is the number of positive perfect squares less than \(n\) and \(g(n)\) is the number of prime numbers less than \(n\). If \(f(x) + g(x) = 16\), then \(x\) is in the range:

A. \(30 \lt x \lt 36\) B. \(30 \lt x \lt 37\) C. \(31 \lt x \lt 37\) D. \(31 \lt x \lt 38\) E. \(32 \lt x \lt 38\)

The functions \(f\) and \(g\) are defined for all the positive integers \(n\) by the following rule: \(f(n)\) is the number of positive perfect squares less than \(n\) and \(g(n)\) is the number of prime numbers less than \(n\). If \(f(x) + g(x) = 16\), then \(x\) is in the range:

A. \(30 \lt x \lt 36\) B. \(30 \lt x \lt 37\) C. \(31 \lt x \lt 37\) D. \(31 \lt x \lt 38\) E. \(32 \lt x \lt 38\)

Hi bunuel I cannot understand why f(31) = 5 ?????? Can you explain it ! tks !

\(f(n)\) is the number of positive perfect squares less than \(n\), so f(31) is the number of positive perfect squares less than 31. There are 5 positive positive perfect squares less than 31: 1, 4, 9, 16, and 25. Therefore, f(31) = 5.

The functions \(f\) and \(g\) are defined for all the positive integers \(n\) by the following rule: \(f(n)\) is the number of positive perfect squares less than \(n\) and \(g(n)\) is the number of primes numbers less than \(n\). If \(f(x) + g(x) = 16\), then \(x\) is in the range:

A. \(30 \lt x \lt 36\) B. \(30 \lt x \lt 37\) C. \(31 \lt x \lt 37\) D. \(31 \lt x \lt 38\) E. \(32 \lt x \lt 38\)

If \(x = 31\), then \(f(31) = 5\) and \(g(31) = 10\): \(f(x) + g(x) = 5 + 10 = 15\).

If \(x = 32\), then \(f(32) = 5\) and \(g(32) = 11\): \(f(x) + g(x) = 5 + 11 = 16\).

...

If \(x = 36\), then \(f(36) = 5\) and \(g(36) = 11\): \(f(x) + g(x) = 5 + 11 = 16\).

If \(x = 37\), then \(f(37) = 6\) and \(g(37) = 11\): \(f(x) + g(x) = 6 + 11 = 17\).

Thus \(x\) could be 32, 33, 34, 35 or 36: \(31 \lt x \lt 37\).

Answer: C

Wouldn't D also be correct, since 31<x<37 would also be included in 31<x<38, and therefore "\(x\) is in the range" 31<x<38?

I think the question should instead be something like "If \(f(x) + g(x) = 16\), then the exact range of possible values of x is:" or "If \(f(x) + g(x) = 16\), then which of the following represents all possible values of x?"

The functions \(f\) and \(g\) are defined for all the positive integers \(n\) by the following rule: \(f(n)\) is the number of positive perfect squares less than \(n\) and \(g(n)\) is the number of primes numbers less than \(n\). If \(f(x) + g(x) = 16\), then \(x\) is in the range:

A. \(30 \lt x \lt 36\) B. \(30 \lt x \lt 37\) C. \(31 \lt x \lt 37\) D. \(31 \lt x \lt 38\) E. \(32 \lt x \lt 38\)

If \(x = 31\), then \(f(31) = 5\) and \(g(31) = 10\): \(f(x) + g(x) = 5 + 10 = 15\).

If \(x = 32\), then \(f(32) = 5\) and \(g(32) = 11\): \(f(x) + g(x) = 5 + 11 = 16\).

...

If \(x = 36\), then \(f(36) = 5\) and \(g(36) = 11\): \(f(x) + g(x) = 5 + 11 = 16\).

If \(x = 37\), then \(f(37) = 6\) and \(g(37) = 11\): \(f(x) + g(x) = 6 + 11 = 17\).

Thus \(x\) could be 32, 33, 34, 35 or 36: \(31 \lt x \lt 37\).

Answer: C

Wouldn't D also be correct, since 31<x<37 would also be included in 31<x<38, and therefore "\(x\) is in the range" 31<x<38?

I think the question should instead be something like "If \(f(x) + g(x) = 16\), then the exact range of possible values of x is:" or "If \(f(x) + g(x) = 16\), then which of the following represents all possible values of x?"

Hello. D can't be the answer because (1) there are 6 perfect squares (1, 4, 9, 16, 25, 36) < 38 (2) there are 11 prime numbers < 38 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37) ==> f(x) + g(x) could be 6 + 11 = 17, NOT 16.

You have to narrow the range in order to satisfy the equation f(x) + g(x) = 16.

Thus, D is wrong.
_________________

Please +1 KUDO if my post helps. Thank you.

"Designing cars consumes you; it has a hold on your spirit which is incredibly powerful. It's not something you can do part time, you have do it with all your heart and soul or you're going to get it wrong."

The functions \(f\) and \(g\) are defined for all the positive integers \(n\) by the following rule: \(f(n)\) is the number of positive perfect squares less than \(n\) and \(g(n)\) is the number of primes numbers less than \(n\). If \(f(x) + g(x) = 16\), then \(x\) is in the range:

A. \(30 \lt x \lt 36\) B. \(30 \lt x \lt 37\) C. \(31 \lt x \lt 37\) D. \(31 \lt x \lt 38\) E. \(32 \lt x \lt 38\)

If \(x = 31\), then \(f(31) = 5\) and \(g(31) = 10\): \(f(x) + g(x) = 5 + 10 = 15\).

If \(x = 32\), then \(f(32) = 5\) and \(g(32) = 11\): \(f(x) + g(x) = 5 + 11 = 16\).

...

If \(x = 36\), then \(f(36) = 5\) and \(g(36) = 11\): \(f(x) + g(x) = 5 + 11 = 16\).

If \(x = 37\), then \(f(37) = 6\) and \(g(37) = 11\): \(f(x) + g(x) = 6 + 11 = 17\).

Thus \(x\) could be 32, 33, 34, 35 or 36: \(31 \lt x \lt 37\).

Answer: C

Wouldn't D also be correct, since 31<x<37 would also be included in 31<x<38, and therefore "\(x\) is in the range" 31<x<38?

I think the question should instead be something like "If \(f(x) + g(x) = 16\), then the exact range of possible values of x is:" or "If \(f(x) + g(x) = 16\), then which of the following represents all possible values of x?"

Hello. D can't be the answer because (1) there are 6 perfect squares (1, 4, 9, 16, 25, 36) < 38 (2) there are 11 prime numbers < 38 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37) ==> f(x) + g(x) could be 6 + 11 = 17, NOT 16.

You have to narrow the range in order to satisfy the equation f(x) + g(x) = 16.

Thus, D is wrong.

I think you misunderstand the unintended meaning of the question and my point. The question tells us that "\(f(x) + g(x) = 16\)" and then asks "\(x\) is in the range:"

That last part is very important. Because it asks "x IS in the range" (emphasis added), we are basically being asked a must be true question. We don't know exactly what x is, but we need to find the answer with a range of values that definitely includes whatever x actually is.

Since we know that the range of possible x values is 32-36, x must be one of those numbers and the correct answer must include that entire range to be correct.

D says \(31 \lt x \lt 38\). While this includes more than all of the possible values of x, it still includes all of the possible values of x. Thus, x is definitely in the range given in D, as directly asked by the question.

I completely understand the math. I completely understand the intent of the question. I got the question correct when I did it, but I noticed that other answers are also technically correct. The way the question is worded does not match the intent of the question and should be fixed.

As a general piece of advice, one should be very careful about what the GMAT asks. I often see test-takers answer a different question than what is asked or not understand how to analyze "must be true" questions.

I think this is a high-quality question and the explanation isn't clear enough, please elaborate. What exactly are we doing here ? Are we plugging values into the function by picking one from the range in the answer choices ?

Indeed, mmagyar is right - the appropriate question for this task is: "In what range all x satisfy f(x)+g(x)=16". Otherwise, the question itself is incorrect, since if I pick one x from a range, and it fits, then I answer the question.

Well, I understand your point of the need for preciseness of wording.

However, choice D and E are incorrect because now E excludes x=32 and D includes x=37.

Even though choice D overlaps with C, like you said, it introduces a new problem. This is crucial point to remember when solving not just in quant but in critical reasoning questions as well.

You must choose an answer that corrects the problem but does not introduces a new problem that compromises the solution.

The wording of the question has absolutely no problem "x is in the range..."