ISEE Upper Level Quantitative : Algebraic Concepts
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Example Questions
Example Question #3 : How To Subtract Variables
When evaluating the expression
assuming you know the values of all five variables, what is the third operation that must be performed?
The middle subtraction
The rightmost addition
The leftmost addition
The rightmost subtraction
The leftmost subtraction
The leftmost subtraction
In the order of operations, any operations in parentheses must be performed first - there are two, the leftmost addition and the middle subtraction. What remains are the leftmost subtraction, the rightmost subtraction, and the rightmost addition. Since additions and subtractions are performed from left to right, the next, or third, operation performed is the leftmost subtraction.
Example Question #1 : How To Add Variables
Simplify:
Group and collect like terms:
Example Question #2 : How To Add Variables
Simplify the expression:
The expression cannot be simplified further.
The expression cannot be simplified further.
There are no like terms among these three; one has only 
Example Question #3 : How To Add Variables
Which is the greater quantity?
(a) The sum of the integers from 1 to 1,000
(b) The sum of the integers from 1,001 to 1,500
(a) and (b) are equal
It is impossible to tell from the information given
(b) is greater
(a) is greater
(b) is greater
(a) The sum of the integers from 1 to 1,000 is
(b) The sum of the integers from 1,001 to 1,500 is
(b) is greater
Example Question #2 : How To Add Variables
Which is the greater quantity?
(A)
(B)
It is impossible to determine which is greater from the information given
(A) is greater
(A) and (B) are equal
(B) is greater
It is impossible to determine which is greater from the information given
Which quantity is the greater depends on the value of 
Case 1:
Then
and
Subsequently, (A) is greater
Case 2:
Then
and
Subsequently, (B) is greater
Example Question #2 : How To Add Variables
Evaluate for 
Substitute and evaluate, remembering that in order of operations, squaring comes before adding:
Example Question #1 : How To Add Variables
Which of the following is the greater quantity?
(A)
(B)
(B) is greater
(A) is greater
(A) and (B) are equal
It is impossible to determine which is greater from the information given
(B) is greater
(A) Substitute and evaluate, remembering that by order of operations, squaring precedes addition:
(B) Substitute and evaluate
(B) is greater
Example Question #3 : How To Add Variables
(A)
(B)
It is impossible to determine which is greater from the information given
(A) is greater
(B) is greater
(A) and (B) are equal
(B) is greater
Since 
Also, 
making (B) greater.
Example Question #206 : Algebraic Concepts
(A)
(B)
It is impossible to determine which is greater from the information given
(A) is greater
(B) is greater
(A) and (B) are equal
It is impossible to determine which is greater from the information given
The values of both terms depend on the value of 
Case 1:
Case 2:
Therefore, the information given is insufficient.
Example Question #3 : How To Add Variables
When evaluating the expression
assuming you know the values of all five variables, what is the third operation that must be performed?
Multiplication
Division
Subtraction
Squaring
Addition
Multiplication
The operation that must be performed first is the operation in parentheses, which is the division.
After this, the order of operations is:
Exponents
Multiplication and division (left to right)
Addition and subtraction (left to right)
The second thing that must be done is the exponentiation - the squaring. Of the three remaining operations, the multiplication takes precedence. The correct response is multiplication.
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All ISEE Upper Level Quantitative Resources
