GMAT Data Sufficiency - Given that (P + 2Q) is a positive number, what is the value of (P + 2Q)?

Question: Given that (P + 2Q) is a positive number, what is the value of (P + 2Q)?

(1) Q = 2
(2) P^2 + 4PQ + 4Q^2 = 28

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“Given that (P + 2Q) is a positive number, what is the value of (P + 2Q)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". The GMAT Quant section consists of a total of 31 questions.
GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions. Data sufficiency questions are usually refined questions for the GMAT. Candidates while solving these questions do not actually understand the hints and find the questions complex.

Solution and Explanation:

Approach Solution 1:

The given question states that (P + 2Q) is a positive number and it asks to and we want to know the value of P. We don’t know that (P + 2Q) is a positive integer, just a positive number of some kind.

The first statement evaluates that Q = 2

Within this statement, the value of only Q has been presented and nothing is stated about the value of P. The statement in itself lacks clarity and does not evaluate the values. Moreover, no unique value is stated about P + 2Q which makes the statement completely insufficient based on the question.

The second condition states that P^2 + 4PQ + 4Q^2 = 28

Now, this may be identified as a pattern-recognition stretch but it can be simply stated as the “Square of a Sum” pattern. In a more precise manner, this can be written as-

P^2 + 2 * P * (2Q) + (2Q)^2 = (P + 2Q)^2 = 28

The above equation can now be considered as the “Square of a Sum” pattern, with P in the role of A and 2Q in the role of B. Now the same equation needs to be evaluated as equal to the square of the sum:

P^2 + 2 * P * (2Q) + (2Q)^2 = (P + 2Q)^2 = 28

Based on the above equation, what needs to be done is take the square root of it. Normally, both the positive and negative square root are needed to be considered to evaluate the statement. However, the given scenario in the question guarantees that (P + 2Q) is a positive number. Accordingly, taking only the positive square root of the equation can be evaluated as-

(P + 2Q) = \(\sqrt{28}\)

Hence, based on the evaluation equated above, the value of (P + 2Q) has been determined as the Square Root of 28. This typically signifies the unique value for (P + 2Q) and hence, the second statement given is correct.

Correct Answer: B

Approach Solution 2:

Based on the given questions scenario where the value of (P + 2Q) needs to be evaluated as an unique value, considering that (P + 2Q) is a positive number. Accordingly, there are two statement conditions given which need to be proved sufficient to get the value of the given question. Each statement can be evaluated as follows:

Statement 1 identifies with Q = 2. Accordingly, it is obviously insufficient because there is no value given about P , so it is not possible to determine the value of (p+2q)

Statement 2 provides the value with P^2 + 4PQ + 4Q^2 = 28 which needs to be evaluated. Accordingly, in order to get the equation from (P + 2Q), it can be raised to power 2. Extending the value of the same by using the power 2, the statement to be evaluated would be-

P^2 + 4 * Q^2 + 4 * P * Q

The second statement further identifies that this equation is equal to 28

So , we have : (P+2Q ) ^2 = 28 or : (p+2Q) = 28 ^(1/2 ).

Accordingly, it has been determined that the the value (P +2Q)^2 = 28 or it can be stated as (P + 2Q) = 28^(1/2)

Hence, statement 2 gives out a unique value determining that it is sufficient to solve the question.

Correct Answer: B

Approach Solution 3:

According on the question's situation, where (P + 2Qvalue )'s must be evaluated as a singular value while taking its positive nature into account. In order to determine the value of the supplied question, two statement requirements must be shown to be sufficient. The following criteria can be applied to each claim:

Statement 1 matches Q = 2, thus. As a result, it is manifestly insufficient because no value is provided for P, making it impossible to calculate the value of (p+2q).

The number in Statement 2 is P2 + 4PQ + 4Q2 = 28, which requires evaluation. As a result, the equation from (P + 2Q) may be obtained by raising it to the power 2. By multiplying the value of the same by 2, the following statement would need to be evaluated:

P^2 + 4 * Q^2 + 4 * P * Q

Second statement confirms that this equation equals 28.

Therefore, we may write (P+2Q) = 28 or (p+2Q) = 28 (1/2).

In light of this, it has been found that the value (P + 2Q)2 = 28 or (P + 2Q) = 28(1/2)

As a result, statement 2 outputs a distinct value, proving that it is adequate to answer the question.

Correct Answer: B

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