All ISEE Middle Level Quantitative Resources
Example Questions
Example Question #11 : Isee Middle Level (Grades 7 8) Quantitative Reasoning
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)
(b) is the greater quantity
(a) and (b) are equal
It is impossible to determine which is greater from the information given
(a) is the greater quantity
(a) and (b) are equal
so
and
The two are equal regardless of the value of .
Example Question #12 : Isee Middle Level (Grades 7 8) Quantitative Reasoning
and are both negative numbers. Which is the greater quantity?
(a)
(b)
(b) is the greater quantity
(a) is the greater quantity
It is impossible to determine which is greater from the information given
(a) and (b) are equal
(a) and (b) are equal
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,
Example Question #13 : Isee Middle Level (Grades 7 8) Quantitative Reasoning
Which is the greater quantity?
(a)
(b) 18
(a) is the greater quantity
It is impossible to determine which is greater from the information given
(a) and (b) are equal
(b) is the greater quantity
It is impossible to determine which is greater from the information given
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.
Example Question #1 : How To Subtract Variables
What is the value of ?
To solve for , the fractions should first be converted to ones that share a common denominator. Given that , the common denominator is 12.
Thus, can be converted to . This gives us:
Example Question #2 : How To Subtract Variables
Simplify:
It is easiest to begin by moving like terms together. Hence:
becomes
(Notice that is its own term.)
Now, consider the coefficients for each term.
For , you have
For , you have
Hence, the expression simplifies to:
This can be moved around to get the correct answer (which means the same thing):
Example Question #2 : How To Subtract Variables
Simplify:
Begin by distributing the two groups. Notice that you must distribute the subtraction through the groups:
becomes
Next, you should move like terms next to each other:
(Notice that is its own term.)
Now, combine terms.
For , you get
For , you get
Therefore, the final form of the expression is:
Example Question #3 : How To Subtract Variables
Solve for :
Begin by distributing. Thus,
becomes
(Don't forget that you have to distribute your subtraction for the second group.)
Combine like terms on the right side of the equation:
Next, move the values to the left side of the equation and all of the other values to the right side:
Combine like terms on the left:
Finally, divide everything by :
This comes out to be:
or
Example Question #4 : How To Subtract Variables
Simplify:
Begin by distributing the multiplied groups:
Next, move all similar factors together:
Now, combine each set of similar factors:
Therefore, our answer is:
Example Question #5 : How To Subtract Variables
Simplify:
This problem is not too difficult. Begin by moving all common terms next to each other:
Next, simplify each group of terms that has the same set of variables:
And do not forget that you are left with as well!
Now, combine all of these:
Example Question #7 : How To Subtract Variables
Simplify:
Begin by moving common factors next to each other. Thus,
becomes
Now, combine each set:
Remember, there still is also.
Therefore, the simplified form of the expression is: