Simplifying Expressions - GRE Quantitative Reasoning
Card 0 of 72
a # b = (a * b) + a
What is 3 # (4 # 1)?
a # b = (a * b) + a
What is 3 # (4 # 1)?
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
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
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If 
and 
, then 
If
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.
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.
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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
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
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_
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_
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Quantitative Comparison
is an integer.
Quantity A: 
Quantity B: 
Quantitative Comparison
Quantity A:
Quantity B:
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.
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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Quantitative Comparison
Quantity A: 
Quantity B: 
Quantitative Comparison
Quantity A:
Quantity B:
(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.
(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.
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Quantitative Comparison
x and y are non-zero integers.
Quantity A: (x – y)2
Quantity B: (x + y)2
Quantitative Comparison
x and y are non-zero integers.
Quantity A: (x – y)2
Quantity B: (x + y)2
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.
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.
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Which is the greater quantity: the median of 5 positive sequential integers or the mean of 5 positive sequential integers?
Which is the greater quantity: the median of 5 positive sequential integers or the mean of 5 positive sequential integers?
If the first integer is 
, then 
This is the same as the median.
If the first integer is
This is the same as the median.
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You are told that 
can be determined from the expression:
Determine whether the absolute value of 
is greater than or less than 2.
You are told that
Determine whether the absolute value of
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.
The expression is simplified as follows:
Since
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Which best describes the relationship between 
and 
if 
?
Which best describes the relationship between
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.
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.
Compare your answer with the correct one above
a # b = (a * b) + a
What is 3 # (4 # 1)?
a # b = (a * b) + a
What is 3 # (4 # 1)?
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
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
Compare your answer with the correct one above
If and , then
If
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.
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.
Compare your answer with the correct one above
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
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
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_
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_
Compare your answer with the correct one above
Quantitative Comparison
is an integer.
Quantity A:
Quantity B:
Quantitative Comparison
Quantity A:
Quantity B:
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.
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.
Compare your answer with the correct one above
Quantitative Comparison
Quantity A:
Quantity B:
Quantitative Comparison
Quantity A:
Quantity B:
(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.
(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.
Compare your answer with the correct one above
Quantitative Comparison
x and y are non-zero integers.
Quantity A: (x – y)2
Quantity B: (x + y)2
Quantitative Comparison
x and y are non-zero integers.
Quantity A: (x – y)2
Quantity B: (x + y)2
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.
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.
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Which is the greater quantity: the median of 5 positive sequential integers or the mean of 5 positive sequential integers?
Which is the greater quantity: the median of 5 positive sequential integers or the mean of 5 positive sequential integers?
If the first integer is , then
This is the same as the median.
If the first integer is
This is the same as the median.
Compare your answer with the correct one above
You are told that can be determined from the expression:
Determine whether the absolute value of is greater than or less than 2.
You are told that
Determine whether the absolute value of
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.
The expression is simplified as follows:
Since
Compare your answer with the correct one above
Which best describes the relationship between and if ?
Which best describes the relationship between
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.
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.
Compare your answer with the correct one above
a # b = (a * b) + a
What is 3 # (4 # 1)?
a # b = (a * b) + a
What is 3 # (4 # 1)?
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
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
Compare your answer with the correct one above
If and , then
If
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.
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.
Compare your answer with the correct one above