Operations - ISEE Middle Level Quantitative Reasoning
Card 0 of 20
Which quantity is greater if 
?
Which quantity is greater if
We know that 
is always positive for all values of 
. Therefore 
would be negative for all values of 
. From this conclusion, we know:
So we have:
is the greater quantity.
We know that
So we have:
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Which quantity is greater if 
?
Which quantity is greater if
A positive number raised to the third power will be positive, while a negative number raised to the third power will remain negative.
If 
, then 
and 
.
If 
, then 
and 
.
Since we do not know if 
is positive or negative, we cannot draw a conclusion about which option is greater.
If 
, then 
is greater.
If 
, then 
is greater.
A positive number raised to the third power will be positive, while a negative number raised to the third power will remain negative.
If
If
Since we do not know if
If
If
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Which quantity is greater if 
?
Which quantity is greater if
When 
we can write:
We know that 
and 
. Based on this, we can compare the two given quantities.
is the greater quantity.
When
We know that
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Which quantity is greater if 
?
Which quantity is greater if
We know that 
is greater than 
. We can easily test a few values for to determine if the values are increasing or decreasing.
If 
:
If 
:
If 
:
The value of 
is increasing, with the smallest possible value being 
. From this, we know that 
, so 
.
We know that
If
If
If
The value of
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Which of the following is equivalent to 
?
Which of the following is equivalent to
Using the distributive property:
and
Using the associative property of multiplication:
We can rewrite 
as 
; using the commutative and associative properties of multiplication:
is the sum of unlike terms and cannot be simplified.
is the correct choice.
Using the distributive property:
and
Using the associative property of multiplication:
We can rewrite
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Which of the following is equivalent to 
?
Which of the following is equivalent to
The expression is the sum of two unlike terms, and therefore cannot be further simplified. None of these responses is correct.
The expression is the sum of two unlike terms, and therefore cannot be further simplified. None of these responses is correct.
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is a positive integer.
Which is the greater quantity?
(A) 
(B) 
Which is the greater quantity?
(A)
(B)
Depending on the value of 
, it is possible for either expression to be greater or for both to be equal.
Case 1: 
and
So the two are equal.
Case 2: 
and
So (B) is greater.
The correct response is that it cannot be determined which is greater.
Depending on the value of
Case 1:
and
So the two are equal.
Case 2:
and
So (B) is greater.
The correct response is that it cannot be determined which is greater.
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is a positive integer.
Which is the greater quantity?
(A) 
(B) 
Which is the greater quantity?
(A)
(B)
Since 
, 
, so (A) is greater regardless of the value of 
.
Since
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is a positive integer.
Which is the greater quantity?
(A) 
(B) 
Which is the greater quantity?
(A)
(B)
Since 
, and 
is positive,
then by the multiplication property of inequality,
making (A) greater regardless of the value of 
.
Since
then by the multiplication property of inequality,
making (A) greater regardless of the value of
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is a positive integer.
Which is the greater quantity?
(A) 
(B) 
Which is the greater quantity?
(A)
(B)
Regardless of the value of 
, the expressions are equal.
Regardless of the value of
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Define an operation on the real numbers as follows:
For all real values of 
and 
,
is a positive number. Which is the greater quantity?
(a) 
(b) 
Define an operation on the real numbers as follows:
For all real values of
(a)
(b)
so
and
The two are equal regardless of the value of 
.
so
and
The two are equal regardless of the value of
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Which is the greater quantity?
(a) 
(b) 18
Which is the greater quantity?
(a)
(b) 18
The information is insufficient, as we see by exploring two cases:
Case 1: 
Case 2: 
Remember, the three variables need not stand for whole numbers.
The information is insufficient, as we see by exploring two cases:
Case 1:
Case 2:
Remember, the three variables need not stand for whole numbers.
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and 
are both negative numbers. Which is the greater quantity?
(a) 
(b) 
(a)
(b)
The two quantities are equal regardless of the values of 
and 
. To see this, we note that
and
Therefore, by the addition property of equality,
The two quantities are equal regardless of the values of
and
Therefore, by the addition property of equality,
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What is the value of 
?
What is the value of
(The numerator and the denominator are both multiplied by 10 in order to convert the fraction to whole numbers.)
Therefore, 30 is the correct answer.
(The numerator and the denominator are both multiplied by 10 in order to convert the fraction to whole numbers.)
Therefore, 30 is the correct answer.
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What is the value of 
?
What is the value of
(The numerator and the denominator are both multiplied by 10 in order to convert the fraction to whole numbers.)
Therefore, 90 is the correct answer.
(The numerator and the denominator are both multiplied by 10 in order to convert the fraction to whole numbers.)
Therefore, 90 is the correct answer.
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Simplify:
Simplify:
Since all of the variables are positive powers, this is easy. Start by reducing the numeric coefficient:
Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:
Since all of the variables are positive powers, this is easy. Start by reducing the numeric coefficient:
Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:
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Simplify:
Simplify:
Since all of the variables are positive powers, this is easy. Start by reducing the numeric coefficient:
Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:
Since all of the variables are positive powers, this is easy. Start by reducing the numeric coefficient:
Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:
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Simplify:
Simplify:
The easiest way to begin with questions like this one is to "flip" the negative exponents to the top or bottom of the fraction. When you do this, you make the exponent's sign positive:
Now, since all of the variables are positive powers, this is easy. Normally, you would begin by reducing the numeric coefficient. This is not necessary since there is only a 5 in the numerator. Therefore, combine the like variables first:
Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:
The easiest way to begin with questions like this one is to "flip" the negative exponents to the top or bottom of the fraction. When you do this, you make the exponent's sign positive:
Now, since all of the variables are positive powers, this is easy. Normally, you would begin by reducing the numeric coefficient. This is not necessary since there is only a 5 in the numerator. Therefore, combine the like variables first:
Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:
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Simplify:
Simplify:
The easiest way to begin with questions like this one is to "flip" the negative exponents to the top or bottom of the fraction. When you do this, you make the exponent's sign positive:
Next, go ahead and reduce the numeric coefficient:
Then, combine the like variables first:
Finally, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:
The easiest way to begin with questions like this one is to "flip" the negative exponents to the top or bottom of the fraction. When you do this, you make the exponent's sign positive:
Next, go ahead and reduce the numeric coefficient:
Then, combine the like variables first:
Finally, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:
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Solve for 
:
Solve for
To start, notice that the left side of the equation has a common factor of 
. You can factor this out:
becomes
Next, you can divide both sides by 
since you know that it does not equal 
:
Simplifying the right side, you get:
Finally, subtract 
from both sides:
To start, notice that the left side of the equation has a common factor of
becomes
Next, you can divide both sides by
Simplifying the right side, you get:
Finally, subtract
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