Data sufficiency problems require a different mindset. It is not “solve this.” Instead, it’s “Is there enough information to answer the question?”

Use these Examples to practice this mindset and time management.

### 1. Yes/No Questions

Is xy < 0?(1) x = y
(2) y = -1(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

Think about what you need to solve the question.
If xy < 0, then just one of x and y can be less than 0.

If x = y, then they both must be negative or positive, meaning it’s not possible for just one to be less than 0. So statement (1) is sufficient to answer the question “no.”

Though y = -1, there is no way to know if x is greater than 0. Statement (2) is insufficient.

The lengths of the sides of a triangle are x, x + 2 and x + 4. Is it possible to find the area of the triangle?

(1) The triangle is a right triangle with a hypotenuse length of 10.
(2) The lengths of two of the sides are 8 and 10.

(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

The tricks here are:

Remember you do not need to do the calculations.
Remember not to carry the information from (1) into (2).
Remember to make sure the answers are specific, not “or” or “sometimes.”

Statement (1)
In a right triangle, two of the sides are the base and the height, and the hypotenuse is the longest side. So Statement (1) tells you that the base is x, the height is x + 2 and the hypotenuse is x + 4. It also tells you that the hypotenuse is 10, so you know x = 6.

Using the value of x you can find the value of the base and height, and therefore find the area. Statement (1) is sufficient.

Statement (2)
Using the information in the question, the lengths of the sides of the triangle can be 6, 8 and 10. This would be a right triangle, so you can find the area. But do not fall into the trap of assuming it is a right triangle because of Statement (1) and the ease of seeing the 3 : 4 : 5 right triangle.

The lengths could also be 8, 10 and 12. The lengths of three sides of a triangle are enough to find the area, so you could find the area of this triangle, too.

BUT there are two possible triangles. You can find the area of either triangle, but don’t know which triangle. So Statement (2) is not sufficient.

Since only Statement (1) is sufficient, the answer is option (A).

If x > 0 and y > 0, is x + y even?(1) x + 3 is a prime number.
(2) y + 1 is a prime number.(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

Think about what you need to solve the question.

For x + y to be even, either both must be even (even + even = even) or both must be odd (odd + odd = even).
All prime numbers are odd, except for 2.
Use both Statements (1) and (2).

If x + 3 is prime, x must be even. (odd + even = odd)
Since x + 3 > 2, all values of x + 3 will be odd.

Though y + 1 is prime, y + 1 can be equal to 2. If y = 1, then though y + 1 is prime, y is odd.

Since y can be even or odd, x + y can be even or odd.

Using both statements, there is not enough information. The answer is option (E).

Note: You can also solve this by using the method of plugging in and seeing what values make x + 3 prime and y + 1 prime.

### 2. Systems of Equations

Solving systems of equations is a skill that the GMAT often uses to create trick questions. A common issue is the number of equations and the number of variables. The algebra chapter in this Guide also reviews this topic.

A student group sold donuts and GMAT books to raise funds. How many GMAT books were sold?(1) 30% of the 90 items sold were GMAT books.
(2) 63 donuts were sold.(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

Statement (1) is sufficient since 30% of 90 is 0.30 × 90 = 27.

Statement (2) is not sufficient since the number of donuts says nothing about how many books were sold. Keep the statements separate. Be careful not to carry the information of 90 items from (1) into (2).

The correct answer is option (A).

What is the value of x?

(1) 3x + 12y = 24
(2) x/4 = 2 – y

(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

You need 2 equations to solve for 2 variables.

So alone, Statement (1) is not sufficient and Statement (2) is not sufficient.

Usually using both statements together would be enough information.

Just to check, simplify both equations.
3x + 12y = 24 → x + 4y = 8
x/4 = 2 – yx = 8 – 4yx + 4y = 8

So Statements (1) and (2) have the same equation. With just one equation, combining the statements is not sufficient. The answer is option (E).

What is the value of xy?
(1) x + y = 8
(2) x – 2y = 2(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

You need 2 equations to solve for 2 variables.

So alone, Statement (1) is not sufficient, and Statement (2) is not sufficient.

Combining the statements gives 2 equations for the 2 variables, so the value can be found.

Note: You should not do the calculations. They are just shown as part of the explanation.

x + y = 8
– (x – 2y = 2)

3y = 6
y = 2

x + y = 8 and y = 2, so x = 6
So xy = 4.

Bob sells twice as many $20 tickets as Tim, and Tim sells three times as many$10 tickets as Bob. How many tickets did Bob sell? Tickets are either $10 or$20.

(1) Tim sold a total of 40 tickets.
(2) Together Bob and Tim sold 70 tickets for $1000. (A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked. (B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked. (C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed. ### Solution You need 2 equations to solve for 2 variables. The equations will be for the total number of tickets and the total amount of money. The question itself gives some information about the total number of tickets. Let x be the number of$20 tickets sold by Tim. Let y be the number of $10 tickets sold by Bob. Bob sold 2x($20 tickets) + y($10 tickets) = 2x + y tickets. Tim sold x($20 tickets) + 3y(\$10 tickets) = x + 3y tickets.

So the total number of tickets is 2x + y + x + 3y = 3x + 4y

Statement (1) gives information for only 1 equation, so is not sufficient.
Tim sold x + 3y = 40 tickets.

Statement (2) gives both the total number of tickets and the amount of money.
Combining the information in the question with statement (2) gives 2 equations for the 2 variables, so the value can be found.

Note: You should not do the calculations. They are just shown as part of the explanation.

number of tickets:
3x + 4y = 70

amount of money:
3x(20) + 4y(10) = 1000
3x + 2y = 50

Subtract the equations:
2y = 20
y = 10

x = 10

Bob sold 30 tickets and Tim sold 40 tickets.

What is the value of x?

(1) x = y – 1
(2) xy = 12

(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

You need 2 equations to solve for 2 variables.

So alone, Statement (1) is not sufficient and Statement (2) is not sufficient.

Using both statements gives 2 equations for the 2 variables. But notice that xy is one of the terms.

If you substitute equation (1) into equation (2), you get a quadratic.
y(y – 1) = 12
y2 – y – 12 = 0
(y – 4)(y + 3) = 0
y = 4 or y = -3

There are 2 possible values for y, so there are 2 values for x.
xy = 12, y = 4, and x = 3
or
y = -3 and x = -4

Since combining the statements is not sufficient, the answer is option (E).

Note: You should not do the calculations. They are just shown as part of the explanation.

What is the value of x?
(1) 2y + 6z = 10
(2) 3x/y + 3z = 6(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

There are 3 variables, so to solve for all 3 variables you would need 3 equations. But the question asks for the value of 1 variable.

Simplify both equations.

2y + 6z = 10 → y + 3z = 5

3x/y + 3z = 6 → 3x = 6(y + 3z) → x = 2(y + 3z)

You see they share the expression y + 3z. This means you could solve for x.
It is not enough information to find y and z, just x.

So alone, Statement (1) is not sufficient and Statement (2) is not sufficient.
Using both statements together is enough information.

Hint: Factoring or simplifying are a good quick check in many data sufficiency problems.

Note: You should not do the calculations. They are just shown as part of the explanation.

Substitute the value from equation (1) into equation (2).
x = 2(y + 3z) = 2(5) = 10

How old is Gloria?
(1) Becky’s age plus twice Alex’s age is four times Gloria’s age.
(2) Alex’s age plus half of Becky’s age is 12.(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

### Solution

Statement (1) is 1 equation with 3 variables, so Statement (1) is not sufficient.

Statement (2) is 1 equation with 2 variables, and neither variable is the value in the question. So Statement (2) is not sufficient.

Combining the two statements, there are 2 equations and 3 variables. It may seem like that is not enough information.
But notice that both statements use Alex’s age and Becky’s age. Write the equation for each statement.

b + 2a = 4g
a + (1/2)b = 12 → 2a + b = 24

Both equations have the same expression, 2a + b. So you can find Gloria’s age using both statements.

The correct answer is choice (C).

Note: You should not do the calculations. They are just shown as part of the explanation.

4g = 2a + b = 24
4g = 24
g = 6
Gloria is 6 years old.