GRE Math : Simplifying Expressions
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Example Questions
Example Question #11 : Simplifying Expressions
a # b = (a * b) + a
What is 3 # (4 # 1)?
8
27
12
20
15
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 7x to 3y
2. Multiply the sum by 4
3. Add x to the product
4. Subtract x – y from the sum
28x + 11y
28x – 13y
28x + 13y
29x + 13y
28x + 12y
28x + 13y
Step 1: 7x + 3y
Step 2: 4 * (7x + 3y) = 28x + 12y
Step 3: 28x + 12y + x = 29x + 12y
Step 4: 29x + 12y – (x – y) = 29x + 12y – x + y = 28x + 13y
Example Question #2 : Simplifying Expressions
Quantitative Comparison
x and y are non-zero integers.
Quantity A: (x – y)2
Quantity B: (x + y)2
The answer cannot be determined from the information given.
Quantity A is greater.
The two quantities are equal.
Quantity B is greater.
The answer cannot be determined from the information given.
Quantity A: (x – y)2 = x2 – 2xy + y2
Quantity B: (x + y)2 = x2 + 2xy + y2
Both have x2 + y2 so cancel those from both columns and just compare –2xy in Quantity A to 2xy in Quantity B. If x = 1 and y = 1, –2xy = –2 and 2xy = 2, so Quantity B is greater. But if x = 1 and y = –1, –2xy = 2 and 2xy = –2, so Quantity A is greater. The contradiction means the answer cannot be determined.
Example Question #3 : How To Simplify An Expression
Which is the greater quantity: the median of 5 positive sequential integers or the mean of 5 positive sequential integers?
The quantities are equal
The relationship cannot be determined
The mean is greater
The median is greater
The quantities are equal
If the first integer is 
This is the same as the median.
Example Question #3 : Simplifying Expressions
You are told that 
Determine whether the absolute value of 
The relationship cannot be determined from the information given.
The quantities are equal
The expression is simplified as follows:
Since 
Example Question #4 : Simplifying Expressions
Quantitative Comparison
Quantity A:
Quantity B:
The relationship cannot be determined from the information given.
Quantity B is greater.
Quantity A is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
(x + y)2 = x2 + 2xy + y2
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 #5 : Simplifying Expressions
Which best describes the relationship between 
The relationship cannot be determined from the information given.
The relationship cannot be determined from the information given.
Use substitution to determine the relationship.
For example, we could plug in 
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, 
This time the first quantity is smaller. Therefore the relationship cannot be determined from the information given.
Example Question #6 : Simplifying Expressions
If 
Cannot be determined
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) + (3x – y + z) = 4x + z
and 5 + 3 = 8
Lets take this equation and multiply the whole thing by 3:
3(4x + z = 8)
Thus, 12x + 3z = 24.
Example Question #7 : Simplifying Expressions
Quantitative Comparison
Quantity A:
Quantity B:
Quantity B is greater.
The two quantities are equal.
Quantity A is greater.
The relationship cannot be determined from the information given.
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. 256x2 = (16x)2 and 49y2 = (7y)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! a2 – b2 = (a + b)(a – b). Thus, if a = 16x and b = 7y, 256x2 – 49y2 = (16x – 7y)(16x + 7y), and the quantities are equal.
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