GMAT Data Sufficiency– Number Properties & Probability

Level of difficulty: Moderate

Question

Set A contains distinct integers: A = {2, 4, 6, -8, x, y}. When two numbers from this set are picked and multiplied, what is the probability that the product is less than zero?

(1) x*y is not equal to zero.

(2) |x| = |y|

Set A contains distinct integers: A = {2, 4, 6, -8, x, y}. When two numbers from this set are picked and multiplied, what is the probability that the product is less than zero?

(1) x*y is not equal to zero.

(2) |x| = |y|

What do we have to find out?

We have to find the probability that the product of two numbers picked from Set A is negative.

Set A contains distinct integers: A = {2, 4, 6, -8, x, y}. When two numbers from this set are picked and multiplied, what is the probability that the product is less than zero?

(1) x*y is not equal to zero.

(2) |x| = |y|

What do we have to find out?

We have to find the probability that the product of two numbers picked from Set A is negative.

What does that mean?

We need to determine one thing1. When will the product of two numbers to be negative?

When will the product of two numbers be negative?The product of two numbers is negative when one of the numbers is positive and the other number is negative.

What is the next step?

We need to determine 1. How many of the given numbers are positive and how many are

negative?

When will the product of two numbers be negative?The product of two numbers is negative when one of the numbers is positive and the other number is negative.

How many of the given numbers are positive and how many are negative?We know for sure that 3 of these numbers are positive

How many of the given numbers are positive and how many are negative?We know for sure that 3 of these numbers are positive

We know that 1 of these numbers is negative

How many of the given numbers are positive and how many are negative?We know for sure that 3 of these numbers are positive

We know that 1 of these numbers is negative

We do not know whether x and y are positive or negative

How many of the given numbers are positive and how many are negative?We know for sure that 3 of these numbers are positive

We know that 1 of these numbers is negative

We do not know whether x and y are positive or negative

Positive Numbers{2, 4, 6}

3 numbers

Negative Numbers{-8}

1 number

Not known{x and y}

2 numbers

How many of the given numbers are positive and how many are negative?We know for sure that 3 of these numbers are positive

We know that 1 of these numbers is negative

We do not know whether x and y are positive or negative

Positive Numbers{2, 4, 6}

3 numbers

Negative Numbers{-8}

1 number

Not known{x and y}

2 numbers

What does the question finally boil down to?

Determining whether x and y are both positive, both negative or one positive and the other negative or both zeroes.

Statement 1: x * y ≠ 0

Let us evaluate the statements

Statement 1: x * y ≠ 0

It is evident that neither x nor y is 0.

Let us evaluate the statements

Statement 1: x * y ≠ 0

It is evident that neither x nor y is 0.

What could x and y therefore be?

• x and y could both be positive• x and y could both be negative• one of x and y could be positive and the other negative

Let us evaluate the statements

Statement 1: x * y ≠ 0

It is evident that neither x nor y is 0.

What could x and y therefore be?

• x and y could both be positive• x and y could both be negative• one of x and y could be positive and the other negative

Let us evaluate the statements

Did we get an answer? Nope. We could not find out how many of the 6 numbers are positive and

how many are negative. Statement 1 is NOT Sufficient

Statement 1: x * y ≠ 0

It is evident that neither x nor y is 0.

What could x and y therefore be?

• x and y could both be positive• x and y could both be negative• one of x and y could be positive and the other negative

Let us evaluate the statements

Did we get an answer? Nope. We do not find out how many of the 6 numbers are positive and how

many are negative. Statement 1 is not Sufficient

What choices can be eliminated Choices A - statement 1 alone is sufficient can be ruled out. Choice D – Each statement is independently sufficient can also be ruled out.

Statement 2: |x| = |y|

Let us evaluate statement 2

Statement 2: |x| = |y|

The magnitude of x is the same as that of y

Let us evaluate statement 2

Statement 2: |x| = |y|

The magnitude of x is the same as that of y

What could x and y therefore be?

• x and y could both be positive. For e.g, x = 7; y = 7

Let us evaluate statement 2

Statement 2: |x| = |y|

The magnitude of x is the same as that of y

What could x and y therefore be?

• x and y could both be positive. For e.g, x = 7; y = 7• x and y could both be negative. For e.g., x = -7; y = -7

Let us evaluate statement 2

Statement 2: |x| = |y|

The magnitude of x is the same as that of y

What could x and y therefore be?

• x and y could both be positive. For e.g, x = 7; y = 7• x and y could both be negative. For e.g., x = -7; y = -7• one of x and y could be positive and the other negative. For

e.g., x = 7; y = -7

Let us evaluate statement 2

Statement 2: |x| = |y|

The magnitude of x is the same as that of y

What could x and y therefore be?

• x and y could both be positive. For e.g., x = 7; y = 7• x and y could both be negative. For e.g., x = -7; y = -7• one of x and y could be positive and the other negative. For

e.g., x = 7; y = -7• Both could be zeroes.

Let us evaluate statement 2

Statement 2: |x| = |y|

The magnitude of x is the same as that of y

What could x and y therefore be?

• x and y could both be positive. For e.g, x = 7; y = 7• x and y could both be negative. For e.g., x = -7; y = -7• one of x and y could be positive and the other negative. For

e.g., x = 7; y = -7• Both could be zero.Before we conclude that statement 2 is not sufficient

Let us evaluate statement 2

Take a look at all the data available to us about these numbers

→ These numbers are all distinct integers

Take a look at all the data available to us about these numbers

→ These numbers are all distinct integers

If x and y have to be distinct, one has to be positive and the other negative.

Take a look at all the data available to us about these numbers

→ These numbers are all distinct integers

If x and y have to be distinct, one has to be positive and the other negative.Therefore, we can eliminate three possibilities.

Take a look at all the data available to us about these numbers

→ These numbers are all distinct integers

If x and y have to be distinct, one has to be positive and the other negative.Therefore, we can eliminate three possibilities.

If x = 7, y has to be -7 or if x = -7, y has to be 7.

Did we get an answer? Yes. 4 of the 6 numbers are positive. 2 are negative Statement 2 is Sufficient

Choice B is the correct Answer.

From a Data Sufficiency point of view what we have done is enough.

Had it been a problem solving question, the required probability is

From a Data Sufficiency point of view what we have done is enough.

Had it been a problem solving question, the required probability is

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