ISEE Upper Level Quantitative : Variables
Study concepts, example questions & explanations for ISEE Upper Level Quantitative
All ISEE Upper Level Quantitative Resources
Example Questions
Example Question #2 : How To Subtract Variables
Simplify:
Example Question #1 : How To Subtract Variables
(A)
(B)
(A) and (B) are equal
(A) is greater
(B) is greater
It is impossible to determine which is greater from the information given
(B) is greater
So by the multiplication property of inequality, when each is multiplied by the negative number 
Also,
so by the addition property of inequality,
or
This makes (B) greater.
Example Question #1 : How To Subtract Variables
Assume you know the values of all four variables in the expression
In which order do you perform the operations in order to evaluate the expression?
Multiply, divide, subtract
Subtract, multiply, divide
Subtract, divide, multiply
Divide, multiply, subtract
Multiply, divide, subtract
Divide, multiply, subtract
Multiplication and division take precedence over subtraction in the order of operations, so these two operations are performed first. The two must be performed from left to right, so the division is worked first, followed by the multiplication. The subtraction is last.
Example Question #192 : Algebraic Concepts
Consider the expression:
Which of the following expressions must be equal in value to the above expression?
I)
II)
III)
III only
I, II, and III
I only
I and III only
I and II only
I and III only
The order of operations is as follows:
Exponents
Multiplication and division (left to right)
Addition and subtraction (left to right)
The expression
is therefore evaluated by multiplying, then dividing, then adding. The net result is that the product 
If we examine (I), we see that, since the multiplication is in parentheses, it is worked first. The division is worked second, then the addition. The order of operations has not changed, so the expressions are equivalent.
If we examine (II), we see that the order of operations has changed so that the addition is worked first. We see through example that the expressions can have different values:
If we examine (III), we see that, since the division is in parentheses, it is worked first. The multiplication is worked second, then the addition. The upshot is the same as in the main expression, however - the product 
The correct response is (I) and (III)
Example Question #2 : 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 rightmost addition
The leftmost subtraction
The rightmost subtraction
The middle subtraction
The leftmost addition
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 #1 : 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 #32 : 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
It is impossible to tell from the information given
(a) is greater
(b) is greater
(a) and (b) are equal
(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 #4 : How To Add Variables
Which is the greater quantity?
(A)
(B)
(A) and (B) are equal
It is impossible to determine which is greater from the information given
(A) is greater
(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 #1 : How To Add Variables
Evaluate for 
Substitute and evaluate, remembering that in order of operations, squaring comes before adding:
All ISEE Upper Level Quantitative Resources
