### All GRE Math Resources

## Example Questions

### Example Question #1 : Simplifying Expressions

a # b = (a * b) + a

What is 3 # (4 # 1)?

**Possible Answers:**

20

15

12

27

8

**Correct answer:**

27

Work from the "inside" outward. Therefore, first solve 4 # 1 by replacing a with 4 and b with 1:

4 # 1 = (4 * 1) + 4 = 4 + 4 = 8

That means: 3 # (4 # 1) = 3 # 8. Solve this now:

3 # 8 = (3 * 8) + 3 = 24 + 3 = 27

### Example Question #1 : Simplifying Expressions

Simplify the result of the following steps, to be completed in order:

1. Add 7*x* to 3*y*

2. Multiply the sum by 4

3. Add *x* to the product

4. Subtract *x –* *y* from the sum

**Possible Answers:**

28*x* + 11*y*

29*x* + 13*y*

28*x* + 12*y*

28*x* + 13*y*

28*x* – 13*y*

**Correct answer:**

28*x* + 13*y*

Step 1: 7*x* + 3*y*

Step 2: 4 * (7*x* + 3*y*) = 28*x* + 12*y*

Step 3: 28*x* + 12*y* + *x* = 29*x* + 12*y*

Step 4: 29*x* + 12*y* – (*x* – *y*) = 29*x* + 12*y* – *x* + *y* = 28*x* + 13*y*

### Example Question #1 : Simplifying Expressions

Quantitative Comparison

*x* and *y* are non-zero integers.

Quantity A: (*x* – *y*)^{2}

Quantity B: (*x* + *y*)^{2}

**Possible Answers:**

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

The answer cannot be determined from the information given.

**Correct answer:**

The answer cannot be determined from the information given.

Quantity A: (*x* – *y*)^{2} = *x*^{2} – 2*xy* + *y*^{2}

Quantity B: (*x* + *y*)^{2} = *x*^{2} + 2*xy* + *y*^{2}

Both have *x*^{2} + *y*^{2} so cancel those from both columns and just compare –2*xy* in Quantity A to 2*xy* in Quantity B. If *x* = 1 and *y* = 1, –2*xy* = –2 and 2*xy* = 2, so Quantity B is greater. But if *x* = 1 and *y* = –1, –2*xy* = 2 and 2*xy* = –2, so Quantity A is greater. The contradiction means the answer cannot be determined.

### Example Question #21 : Simplifying Expressions

Which is the greater quantity: the median of 5 positive sequential integers or the mean of 5 positive sequential integers?

**Possible Answers:**

The quantities are equal

The relationship cannot be determined

The mean is greater

The median is greater

**Correct answer:**

The quantities are equal

If the first integer is , then

This is the same as the median.

### Example Question #1 : How To Simplify An Expression

You are told that can be determined from the expression:

Determine whether the absolute value of is greater than or less than 2.

**Possible Answers:**

The relationship cannot be determined from the information given.

The quantities are equal

**Correct answer:**

The expression is simplified as follows:

Since the value of must be slightly greater for it to be 17 when raised to the 4th power.

### Example Question #1 : Simplifying Expressions

Quantitative Comparison

Quantity A:

Quantity B:

**Possible Answers:**

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

The relationship cannot be determined from the information given.

**Correct answer:**

The relationship cannot be determined from the information given.

(*x* + *y*)^{2} = *x*^{2} + 2*xy* + *y*^{2}

Now, since there are no specifications on what x and y can equal, one or both of them could be 0, making the two columns equal. Any value other than 0 will make the columns unequal because of the additional 2xy term, so the answer cannot be determined.

### Example Question #1 : Simplifying Expressions

Which best describes the relationship between and if ?

**Possible Answers:**

The relationship cannot be determined from the information given.

**Correct answer:**

The relationship cannot be determined from the information given.

Use substitution to determine the relationship.

For example, we could plug in and .

So far it looks like the first expression is greater, but it's a good idea to try other values of x and y to be sure. This time, we'll try some negative values, say, and .

This time the first quantity is smaller. Therefore the relationship cannot be determined from the information given.

### Example Question #1 : Simplifying Expressions

If and , then

**Possible Answers:**

Cannot be determined

**Correct answer:**

We have three variables and only two equations, so we will not be able to solve for each independent variable. We need to think of another solution.

Notice what happens if we line up the two equations and add them together.

*(x* + *y*) + (3*x –* *y* + *z) *= 4x + z

and 5 + 3 = 8

Lets take this equation and multiply the whole thing by 3:

3(4*x* + *z* = 8)

Thus, 12*x* + 3*z* = 24.

### Example Question #2 : Simplifying Expressions

Quantitative Comparison

is an integer.

Quantity A:

Quantity B:

**Possible Answers:**

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

**Correct answer:**

The two quantities are equal.

Plugging in numbers is not the best strategy here. Instead, let's see how we can equate the two expressions. Quantity A is actually a difference of squares. 256*x*^{2} = (16*x*)^{2} and 49*y*^{2} = (7*y*)^{2}. These look like the expressions in Quantity B. The formula to remember here is the difference of squares formula, a very important one for this test! *a*^{2} – *b*^{2} = (*a* + *b*)(*a* – *b*). Thus, if *a* = 16*x* and *b* = 7*y*, 256*x*^{2} – 49*y*^{2 }= (16*x* – 7*y*)(16*x* + 7*y*), and the quantities are equal.

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