Test: Positive & Negative Numbers - GMAT MCQ

Step 1: Decode the Question Stem

Q1. What kind of an answer will the question fetch?
The question is an "IS" question. For "is" questions, the answer is "YES" or "NO".

Step 2: Evaluate Statement 1 ALONE

Statement 1: a - b > 0.

From this statement we can conclude that a > b. But we cannot gain any insight about whether 'a' is positive.
Here are two possible scenarios where the statement is true without helping us arrive at any conclusion.

Example: Both 'a' and 'b' could be negative and 'a' could be greater than 'b'.
For example, a = -5 and b = -10. a - b > 0. 'a' is negative.

Counter Example: Alternatively 'a' could be positive.
For example, a = 10 and b could be 3. a - b > 0 and 'a' is positive.
A counter example exists.

Statement 1 alone is NOT sufficient.
Eliminate answer options A and D.

Step 3: Evaluate Statement 2 ALONE

Statement 2: 2a - b > 0.

From this statement we can conclude that 2a > b. However, we cannot gain any insight about whether 'a' is positive.
Let us check out the following two scenarios.

Example: Let a = -3, b = -100, 2a = -6. 2a > b and 'a' is negative.
Counter Example: Let a = 10, b = 12. Therefore, 2a = 20. 2a > b and a is positive.

A counter example exists.
Statement 2 ALONE is NOT sufficient.
Eliminate option B. Choices narrow down to C or E.

Step 4: Evaluate statements together

Statement 1: a - b > 0.
Statement 2: 2a - b > 0.

Let us look at the following two examples.
Example: a = -3, b = -100 and 2a = -6. a > b and 2a > b. 'a' is negative.
Counter Example: a = 20, b = 15 and 2a = 40. a > b and 2a > b. However, 'a' is positive.
A counter example exists.
Even after combining the information in the two statements, we cannot conclude whether 'a' is positive.

Statements together are not sufficient.
Eliminate answer option C.

Choice E is the correct answer.

Step 1 of solving this GMAT DS question: Understand the Question Stem

What kind of an answer will the question fetch?
The question is a "How many?" question. For questions asking "how many", the answer should be a number.

When is the data sufficient?
The data is sufficient if we are able to get a UNIQUE answer for the number of positive numbers from the information in the statements.
If the statements do not have adequate data to uniquely determine how many among the three numbers are positive, the data is NOT sufficient.

Key data from the question stem
Each of the three numbers x, y, and z are less than 10.

Step 2 of solving this GMAT DS question:
Evaluate Statement (1) ALONE: x + y + z = 20

From the question stem we know that each number is less than 10.
So, x < 10, y < 10 and z < 10.
Therefore, the maximum sum of any two of these numbers, say x + y < 20.

However, statement 1 states x + y + z = 20.
Unless the third number, z in this case, is also positive x + y + z cannot be 20.
Hence, we can conclude that all 3 numbers x, y and z are positive.

Statement 1 ALONE is sufficient.
Eliminate choices B, C, and E. Choices narrow down to A or D.

Step 3 of solving this GMAT DS question:
Evaluate Statement (2) ALONE: x + y = 14

As each of x and y is less than 10, both x and y have to be positive for the sum to be 14.
However, z could also be positive or z could be negative.

So, there could be either 2 or 3 positive numbers among the three numbers.
We are not able to find a unique answer from the information in statement 2.

Statement 2 ALONE is NOT sufficient.
Eliminate choice D.

Statement 1 ALONE is sufficient. Choice A is the answer.

We are told that a/b is negative. We don't know whether a in particular is positive or negative, but we can analyze the situation by cases. If a is positive, then b must be negative, in order for a/b to be negative. If a is negative, then b must be positive, in order for a/b to be negative. Similar cases hold when we start with b and consider a: one must be positive and one negative. This means that (C) cannot be true, because ab would only be positive if both numbers were positive or both numbers were negative.
The correct answer is (C).

Step 1 of solving this GMAT DS question: Understand the Question Stem

What kind of an answer will the question fetch?
The question is an "Is" question. Answer to an "is" questions is either YES or NO.

When is the data sufficient?
The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

Step 2 of solving this GMAT DS question:
Evaluate Statement (1) ALONE: (a + b)² < (a - b)²

Expanding both sides of the inequality, we get a² + b² + 2ab < a² + b² - 2ab
Simplifying we get, 4ab < 0 or ab < 0.
So, we can conclude that ab is not positive. We have got a definite NO as the answer.

Statement 1 ALONE is sufficient.
Eliminate choices B, C and E. Choices narrow down to A or D.

Step 3 of solving this GMAT DS question:
Evaluate Statement (2) ALONE: a = b

This is actually the statement that could trick you.
a = b.
So, either both a and b or positive or both a and b are negative. In either case ab is positive.
We will certainly be "tempted" to decide that statement 2 is also sufficient.

The catch is that, both a and b could be 0. In that case ab = 0, which is not positive.
If 'a' and 'b' are either both positive or both negative, the answer is yes. If both are 0, the answer is no.
As we are not able to conclude whether ab is positive with statement 2, it is not sufficient.

Statement 2 ALONE is NOT sufficient.
Eliminate choice D.

Statement 1 ALONE is sufficient but statement 2 is not sufficient. Choice A is the answer.

Step 1 of solving this GMAT DS question: Understand the Question Stem

What kind of an answer will the question fetch?
The question is an "Is" question. Answer to an "is" questions is either YES or NO.

When is the data sufficient?
The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

What additional information do we have about P from the question stem?
'P' is a 2-digit positive integer.

Step 2 of solving this GMAT DS question:
Evaluate Statement (1) ALONE: (P + 2) and (P - 2) are prime.

Inference: (P - 2), P and (P + 2) are 3 consecutive odd integers.
Why?
Because (P - 2) and (P + 2) are prime, both numbers have to be odd.
(P - 2), P, and (P + 2) are three numbers in an arithmetic progression with a common difference of 2.
So, the 3 numbers have to be 3 consecutive odd or consecutive even integers. If (P - 2) and (P + 2) are odd, then these 3 numbers have to be 3 consecutive odd integers.

One out of 3 consecutive odd integers, (P - 2), P, and (P + 2) will definitely be a multiple of '3'.
If (P + 2) and (P - 2) are prime, then P has to be a multiple of '3', which is not prime.

The only exception is if the 3 consecutive odd numbers are 3, 5, and 7. However, we are dealing with two digit positive integers. So that possibility is ruled out.

Statement 1 ALONE is sufficient.
Eliminate choices B, C, and E. Choices narrow down to A or D.

Step 3 of solving this GMAT DS question:
Evaluate Statement (2) ALONE: (P - 4) and (P + 4) are prime.

This is a brilliant statement.
1. The remainder when (P - 4) and (P - 1) are divided by 3 will be the same.
2. Similarly, the remainder when (P + 4) and (P + 1) are divided by 3 will be the same.
If (P - 4) and (P + 4) are prime, both (P - 4) and (P + 4) will leave a remainder when divided by 3.

Therefore, (P - 1) and (P + 1) will also leave a remainder when divided by 3. i.e., they are not divisible by 3.
(P - 1), P, (P + 1) are 3 consecutive positive integers.
One out of 3 consecutive integers, (P - 1), P, and (P + 1) will definitely be a multiple of '3'.
If (P - 1) and (P + 1) are not divisible by 3, then P has to be a multiple of '3'.
P cannot be 3 because P is a 2-digit number. So, that possiblity is ruled out.
Any 2-digit number that is a multiple of 3 cannot be prime.
Therefore, P is not prime.

Statement 2 ALONE is also sufficient.
Eliminate choice A.

Each statement is INDEPENDENTLY sufficient. Choice D is the answer.