All ISEE Middle Level Quantitative Resources
Example Questions
Example Question #1 : How To Add Variables
Which quantity is greater if ?
and
are equal
It is not possible to tell from the information given
is greater
is greater
is greater
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.
Example Question #1 : How To Add Variables
Which quantity is greater if ?
It is not possible to tell from the information given
is greater
is greater
and
are equal
It is not possible to tell from the information given
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.
Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning
Which quantity is greater if ?
is greater
and
are equal
It is not possible to tell from the information given
is greater
is greater
When we can write:
We know that and
. Based on this, we can compare the two given quantities.
is the greater quantity.
Example Question #1 : How To Add Variables
Which quantity is greater if ?
and
are equal
is greater
It is not possible to tell from the information given
is greater
is greater
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
.
Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning
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.
Example Question #1 : How To Add Variables
is a positive integer.
Which is the greater quantity?
(A)
(B)
(A) and (B) are equal
It is impossible to determine which is greater from the information given
(B) is greater
(A) is greater
It is impossible to determine which is greater from the information given
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.
Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning
is a positive integer.
Which is the greater quantity?
(A)
(B)
(A) is greater
(B) is greater
It is impossible to determine which is greater from the information given
(A) and (B) are equal
(A) is greater
Since , and
is positive,
then by the multiplication property of inequality,
making (A) greater regardless of the value of .
Example Question #1 : How To Add Variables
is a positive integer.
Which is the greater quantity?
(A)
(B)
(A) and (B) are equal
(B) is greater
It is impossible to determine which is greater from the information given
(A) is greater
(A) and (B) are equal
Regardless of the value of , the expressions are equal.
Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning
Which of the following is equivalent to ?
None of the other responses is correct.
None of the other responses is correct.
The expression is the sum of two unlike terms, and therefore cannot be further simplified. None of these responses is correct.
Example Question #2 : How To Add Variables
is a positive integer.
Which is the greater quantity?
(A)
(B)
(A) and (B) are equal
(A) is greater
It is impossible to determine which is greater from the information given
(B) is greater
(A) is greater
Since ,
, so (A) is greater regardless of the value of
.
Certified Tutor
All ISEE Middle Level Quantitative Resources
![Learning Tools by Varsity Tutors](https://vt-vtwa-app-assets.varsitytutors.com/assets/problems/og_image_practice_problems-9cd7cd1b01009043c4576617bc620d0d5f9d58294f59b6d6556fd8365f7440cf.jpg)