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QUESTION: 1

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

What is the value of X, if X and Y are two distinct integers and their product is 30?

- X is an odd integer
- X > Y

Solution:

Values of X and Y that satisfy both the conditions are

More than one value exists for X. Because we are not able to deduce a UNIQUE value for X using the information provided in the two statements together, the given data is NOT sufficient.

Statements TOGETHER are NOT sufficient.

Choice E is the answer.

QUESTION: 2

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

What is the standard deviation (SD) of the four numbers p, q, r, s?

- The sum of p, q, r and s is 24
- The sum of the squares of p, q, r and s is 224

Solution:

From statement 1 we know that the square of the means is 36.

From statement 2 we know that the mean of the squares is 56.

Using the formula, Standard deviation =

we can find the SD of the 4 numbers

Statements TOGETHER are SUFFICIENT.

Choice C is the answer.

QUESTION: 3

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

How is Bill related to Betty?

1. Cindy, the wife of Bill's only brother Chris does not have any siblings.

2. Betty is Cindy's brother in law's wife.

Solution:

We combine the two statements, we know that Bill and Cindy are related to each other through Chris, who is the only brother of Bill and that Cindy is Betty's brother in law's wife.

Cindy does not have any siblings and hence her brother in law has to necessarily be her husband's sibling. As Chris is the only brother of Bill, Cindy's brother in law has to be Bill and Betty is his wife.

Statements TOGETHER are SUFFICIENT.

Choice C is the answer.

QUESTION: 4

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

Is y an integer?

1. y^{3} is an integer

2. 3y is an integer

Solution:

**Combine the two statements**: We know that y^{3} is an integer and 3y is also an integer.

Only for integer values of y, will both y^{3} and 3y be integers simultaneously.

Statements TOGETHER are SUFFICIENT.

Choice C is the correct answer.

QUESTION: 5

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

If a salesman received a commission of 3% of the sales that he has booked in a month, what was the sales booked by the salesman in the month of November 2003?

1. The sales booked by the salesman in the month of November 2003 minus salesman's commission was $245,000

2. The selling price of the sales booked by the salesman in the month of November 2003 were 125 percent of the original purchase price of $225,000.

Solution:

From statement 1, we know the sales value after the salesman's commission is subtracted.

From the question stem, we know his commission is 3% of the sales booked. Then value of sales after subtracting his commission is 100 - 3 = 97% of the sales booked.

Putting the two together, we can deduce that 97% of sales booked = $245,000. So we can find out the sales booked.

Caveat: Do not waste time computing the value. All we need to know is whether the answer will be unique. We know the answer is unique.

Statement 1 ALONE is SUFFICIENT.

Eliminate choices B, C and E.

====

From statement 2, we know that the original purchase of the products is $225,000.

We can compute the sales booked as 125% of 225,000 = 1.25 * 225,000.

Caveat: Do not waste time computing the value. All we need to know is whether the answer will be unique. We know the answer is unique.

Statement 2 ALONE is SUFFICIENT.

Each statement is INDEPENDENTLY sufficient. So, choice D is the correct answer to this data sufficiency question.

QUESTION: 6

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

Is the positive integer m divisible by 6?

1. m is divisible by 3

2. m is divisible by 4

Solution:

Evaluate Statement (1) ALONE: x is divisible by 6

Approach: Look for a counter example

Example: x = 6. It is divisible by 6. However, it is NOT divisible by 12.

Counter Example: x = 12. It is divisible by 6. It is divisible by 12 as well.

Knowing that x is divisible by 6 is not enough to answer the question.

If x is divisible by 6, we can infer that it is divisible by 3 and 2. But we cannot deduce whether it is also divisible by 22 - which is essential to deduce that x is divisible by 12.

**tatement 1 ALONE is NOT sufficient.**

Eliminate choices A and D. Choices narrow down to B, C or E.

Evaluate Statement (2) ALONE: x is divisible by 8

If x is divisible by 8, then x will definitely be divisible by 4.

However, from statement (2) alone we do not know if x is divisible by 3.

Alternative Approach: Look for a counter example

Example: x = 8. It is divisible by 8. However, it is NOT divisible by 12.

Counter Example:x = 24. It is divisible by 8. It is divisible by 12 as well.

Knowing that x is divisible by 8 is not enough to answer the question.

**Statement 2 ALONE is NOT sufficient.**

Eliminate choice B. Choices narrow down to C or E.

Evaluate Statements (1) & (2) Together: x is divisible by 6 & x is divisible by 8

From statement 1, if x is divisible by 6, it is definitely divisible by 3.

From statement 2, if x is divisible by 8, it is definitely divisible by 4.

So, by combining the two statements, we can conclude that x is divisible by 3 and by 4.

Or that x is divisible by 12.

QUESTION: 7

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

Is ab positive?

1. (a + b)^{2} < (a - b)^{2
2. a = b}

Solution:

STATEMENT 1:

Expanding both sides of the inequality, we get a^{2} + b^{2} + 2ab < a^{2} + b^{2} - 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. Answer choices narrow down to A or D.

STATEMENT 2:

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.

As we are not able to conclude whether ab is positive with statement 2, it is not sufficient. Statement 2 ALONE is NOT sufficient.

QUESTION: 8

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

When Y is divided by 2, is the remainder 1?

1. (-1) ^{(Y+2)} = -1

2. Y is prime.

Solution:

STATEMENT 1:

(-1)^{ODD NUMBER} = -1

Therefore, Y + 2 is an odd number.

Hence, Y has to be an odd number.

So, when Y is divided by 2, the remainder is 1.

Statement 1 ALONE is SUFFICIENT.

Eliminate choices B, C, and E. Answer choices narrow down to A or D.

STATEMENT 2:

Y could be '2' which is an even number.

So, if Y is 2, when Y is divided by 2, the remainder is '0'.

All other prime numbers are odd numbers.

So, if Y is one of the other prime numbers, when Y is divided by 2, the remainder is '1'.

We donot have enough data in the question stem or statement 2 to conclude whether Y is 2 or one of the other prime numbers.

As we are not able to conclude whether Y is an even number using statement 2, it is not sufficient.

Statement 2 ALONE is NOT sufficient.

Choice A is the correct answer.

QUESTION: 9

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

Is the two digit positive integer P a prime number?

1. (P + 2) and (P - 2) are prime.

2. (P - 4) and (P + 4) are prime

Solution:

STATEMENT 1:

(P - 2), P and (P + 2) are 3 consecutive odd integers because (P - 2) and (P + 2) are prime.

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.

STATEMENT 2:

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 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 when P is 3, (P - 4) will not be prime. So, that possiblity is ruled out.

Therefore, P is not prime.

Statement 2 ALONE is SUFFICIENT.

QUESTION: 10

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

If m, s are the average and standard deviation of integers a, b, c, and d, is s > 0?

1. m > a

2. a + b + c + d = 0

Solution:

STATEMENT 1:

If a = b = c = d, the average m will be the same as a.

Since m > a, all the elements in the set cannot be the same, and therefore, s > 0.

Statement 1 ALONE is sufficient.

Eliminate choices B, C and E. Choices narrow down to A and D.

STATEMENT 2:

Approach: Look for a counter example

Example: When a = b = c = d = 0, s = 0

Counter Example: When a = -4, b = 0, c = 0, and d = 4, s > 0

Statement 2 ALONE is NOT sufficient.

Eliminate choice D. Choice A is the answer.

QUESTION: 11

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

Is x^{3} > x^{2} ?

1. x > 0

2. x < 1

Solution:

Statement 1: x > 0

We know that x is a positive number.

Interval 1: If 0 < x < 1, then x3 < x^{2}.

For example, (0.5)3 = 0.125, which is lesser than (0.5)2 = 0.25

The answer to the question is NO.

Interval 2: If x > 1, then x^{3} > x^{2}

For example, 23 = 8 which is greater than 22 = 4

The answer to the question is YES.

We do NOT have a DEFINITE answer using statement 1.

Statement 1 ALONE is NOT sufficient.

Eliminate choices A and D. Choices narrow down to B, C or E.

Evaluate Statement (2) ALONE

Statement 2: x < 1

Interval 1: For positive values of x, i.e., 0 < x < 1, we know x^{3} < x^{2}.

The answer to the question is NO.

Interval 2:For negative values of x, x^{3} will be a negative number and x^{2} will be a positive number.

Hence, x^{3} < x^{2}

The answer to the question is NO.

Lastly, what is the answer if x = 0?

When x = 0, x^{3} = x^{2}.

The answer to the question is NO.

Hence, if we know that x < 1, we can conclude that x^{3} is NOT GREATER THAN x^{2}.

We have a DEFINITE answer, even if it is NO.

Statement 2 ALONE is sufficient. Eliminate choices C and E.

Choice B is the answer.

QUESTION: 12

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

Is y = 3?

1. (y - 3)(x - 4) = 0

2. (x - 4) = 0

Solution:

Evaluate Statement (1) ALONE: (y - 3)(x - 4) = 0

If product of the two terms (y - 3) and (x - 4) is 0, then at least one of the two terms equals 0.

(y - 3) = 0 or (x - 4) = 0 or both (y - 3) and (x - 4) equal 0.

i.e., either y = 3 or x = 4 or that both y = 3 and x = 4.

If x = 4, y could take any value. The value of 'y' could be 3 or it could be some other value and the product will still be a 0.

Example: x = 4 and y = 5. The equation holds good. y ≠ 3.

Counter example: x = 4 and y = 3. The equation holds good. y = 3

We CANNOT determine whether 'y' is 3 from this statement.

Statement 1 ALONE is NOT sufficient.

Eliminate choices A and D. Choices narrow down to B, C or E

Evaluate Statement (2) ALONE: (x - 4) = 0

The statement provides no information about y.

Statement 2 ALONE is NOT sufficient.

Eliminate choice B. Choices narrow down to C or E.

Evaluate Statements (1) & (2) Together: (y - 3)(x - 4) = 0 & (x - 4) = 0

When x = 4, (y - 3)(x - 4) will be 0 irrespective of the value that y takes.

Can 'y' be 3? Yes 'y' can be 3.

Is y = 3? Not necessary.It can take values other than 3 and the data in the two statements will still hold good.

Eliminate choice C.

Statements TOGETHER are NOT sufficient. Choice E is the answer.

QUESTION: 13

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

Is x/y a terminating decimal?

1. x is a multiple of 2

2. y is a multiple of 3

Solution:

Evaluate Statement (1) ALONE

Statement (1) : x is a multiple of 2

No information about y has been provided.

Approach: Let us look for a counter example.

Example: When x = 2 and y = 3, x/y is non-terminating.

The answer to the question is NO.

Counter Example: When x = 2 and y = 4, x/y is terminating.

The answer to the question is YES.

We have found a counter example. Therefore, statement 1 does not provide a DEFINITE answer.

**Statement 1 ALONE is NOT sufficient.**

Eliminate choices A and D. Choices narrow down to B, C or E.

Evaluate Statement (2) ALONE

Statement (2) : y is a multiple of 3

No information about x has been provided.

Approach: Let us look for a counter example.

Example:When x = 3 and y = 3, xyxy is terminating.

The answer to the question is YES.

Counter Example: When x = 2 and y = 3, xyxy is non-terminating.

The answer to the question is NO.

We have found a counter example. Therefore, statement 2 does not provide a DEFINITE answer.

**Statement 2 ALONE is NOT sufficient.**

Eliminate choice B. Choices narrow down to C or E.

Evaluate Statements (1) & (2) Together

Statements Together : x is a multiple of 2 & y is a multiple of 3

Approach: Let us look for a counter example.

Example:When x = 6 and y = 6, x/y is terminating.

The answer to the question is YES.

Counter Example: When x = 2 and y = 3, x/y is non-terminating.

The answer to the question is NO.

We have found a counter example. Despite combining the information in the statements we are not able to find a DEFINITE answer.

Statements TOGETHER are NOT SUFFICIENT. Choice E is the answer.

QUESTION: 14

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

Is the positive integer X divisible by 21?

1. When X is divided by 14, the remainder is 4

2. When X is divided by 15, the remainder is 5

Solution:

Evaluating Statement (1) ALONE

Statement (1) : When X is divided by 14, the remainder is 4

The number is, therefore, of the form 14k + 4.

It will leave a remainder of 4 when divided by 7. (14k is divisible by 7. When 4 is divided by 7, the remainder is 4.)

This number is definitely not divisible by 7.

To be divisible by 21, the number must be divisible by both 3 and 7. This number is not divisible by 7. Hence, X is not divisible by 21.

We have a DEFINITE NO as the answer to the question using statement 1.

Statement 1 ALONE is sufficient.

Eliminate choices B, C and E. Choices narrow down to A or D.

Evaluating Statement (2) ALONE

Statement (2) : When X is divided by 15, the remainder is 5

The number X is of the form 15m + 5

Therefore, the number will leave a remainder of 2 when divided by 3.

Hence, it is not divisible by 3.

To be divisible by 21, the number must be divisible by both 3 and 7. This number is not divisible by 3. Hence, X is not divisible by 21.

We have a DEFINITE NO as the answer to the question using statement 2 as well.

Statement 2 ALONE is also sufficient.

Eliminate choice A.

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

QUESTION: 15

**Numbers**

All numbers used are real numbers.

**Figures**

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Q.

A set S contains the following elements: {7, 11, 15, 19, 23, x}. What is the value of x?

1. The elements are in arithmetic progression.

2. x is prime.

Solution:

STATEMENT 1:

The common difference of the sequence is 4.

So, x could either be 3 or 27.

We are not able to find a UNIQUE value for x from statement 1.

Statement 1 ALONE is NOT sufficient.

Eliminate choices A and D. Choices narrow down to B, C or E.

STATEMENT 2:

x could be any prime number. There are infinite possibilities.

We are not able to find a UNIQUE value for x from statement 1.

Statement 2 ALONE is NOT sufficient.

Eliminate choice B. Choices narrow down to C or E

From statement 1 we have narrowed down the values that could take to 3 or 27;

From statement 2 we know that x is prime.

3 is the only value that satisfies both the conditions.

Statements TOGETHER are SUFFICIENT.

Choice C is the answer.

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