All ISEE Upper Level Quantitative Resources
Example Questions
Example Question #15 : How To Find The Solution To An Equation
Ā refers to the greatest integer less than or equal toĀ
.
Ā andĀ
Ā are integers. Which is greater?
(a)Ā
(b)Ā
(a) and (b) are equal
(a)Ā is greater
It is impossible to tell from the information given
(b) is greater
(a) and (b) are equal
If Ā is an integer, thenĀ
Ā by definition.
SinceĀ , and, by closure,Ā
Ā are all integers,Ā
Ā andĀ
, making (a) and (b) equal.
Example Question #15 : Algebraic Concepts
Consider the line of the equationĀ .
Which is the greater quantity?
(a)Ā TheĀ -coordinate of theĀ
-intercept.
(b)Ā TheĀ -coordinate of theĀ
-intercept.
(b) is greater.
(a) and (b) are equal.
It is impossible to tell from the information given.
(a) is greater.
(a) is greater.
(a) To find theĀ -coordinate of theĀ
-intercept, substituteĀ
:
(b)Ā To find theĀ -coordinate of theĀ
-intercept, substituteĀ
:
Therefore (a) is the greater quantity.
Example Question #691 : Isee Upper Level (Grades 9 12) Quantitative Reasoning
Which is the greater quantity?
(a)Ā
(b)Ā
It is impossible to tell from the information given.
(b) is greater.
(a) is greater.
(a) and (b) are equal.
It is impossible to tell from the information given.
Each can be rewritten as a compound statement. Solve separately:
orĀ
Ā
Similarly:
Ā
Ā
Therefore, it cannot be determined with certainty which ofĀ Ā andĀ
Ā is the greater.
Example Question #16 : How To Find The Solution To An Equation
Which is the greater quantity?
(a)Ā
(b)
It is impossible to tell from the information given.
(b) is greater.
(a) and (b) are equal.
(a) is greater.
It is impossible to tell from the information given.
If , then either
Ā or Ā
. Solve for
Ā in both equations:
orĀ
Therefore, either (a) and (b) are equal or (b) is the greater quantity, but it cannot be determined with certainty.
Example Question #691 : Isee Upper Level (Grades 9 12) Quantitative Reasoning
Which is the greater quantity?
(a)Ā
(b)
(a) is greater
(b) is greater
(a) and (b) are equal
It is impossible to tell from the information given
(a) and (b) are equal
Example Question #12 : Algebraic Concepts
Consider the line of the equationĀ .
Which is the greater quantity?
(a)Ā TheĀ -coordinate of theĀ
-intercept
(b)Ā TheĀ -coordinate of theĀ
-intercept
(a) and (b) are equal.
(a) is greater.
It is impossible to tell from the information given.
(b) is greater.
(a) is greater.
(a) To find theĀ -coordinate of theĀ
-intercept, substituteĀ
:
(b)Ā To find theĀ -coordinate of theĀ
-intercept, substituteĀ
:
(a) is the greater quantity.
Example Question #692 : Isee Upper Level (Grades 9 12) Quantitative Reasoning
Ā refers to the least integer greater than or equal toĀ
.
Ā andĀ
Ā are integers.Ā
Which is the greater quantity?
(a)Ā
(b)Ā
(a)Ā is greater.
It is impossible to tell from the information given.
(a) and (b) are equal.
(b) is greater.
(a)Ā is greater.
(a) SinceĀ Ā is an integer,Ā
.
SinceĀ Ā is an integer,Ā
.
(b) By closure,Ā Ā is an integer, soĀ
.
(a) is the greater quantity.
Example Question #22 : How To Find The Solution To An Equation
Ā refers to the greatest integer less than or equal toĀ
.
Ā andĀ
Ā are integers.
Which is greater?
(a)Ā
(b)Ā
(a)Ā is greater.
(a) and (b) are equal.
(b) is greater.
It is impossible to tell from the information given.
(b) is greater.
(a) SinceĀ Ā is an integer,Ā
.
SinceĀ Ā is an integer,Ā
.
(b) By closure,Ā Ā is an integer, soĀ
.
This makes (b) greater.
Example Question #21 : Equations
Ā
Which is the greater quantity?
(a)Ā
(b)
It cannot be determined from the information given.
(a) is greater.
(a) and (b) are equal.
(b) is greater.
(a) and (b) are equal.
SubstituteĀ Ā and, subsequently,Ā
:
Factor asĀ , replacing the two question marks with integers whose product isĀ
Ā and whose sum is
. These integers areĀ
.
Break this up into two equations, replacingĀ Ā forĀ
:
or
This has no solution, sinceĀ Ā must be nonnegative.
is the only solution, so (a) and (b) must be equal.
Ā
Example Question #21 : Algebraic Concepts
Consider the line through pointsĀ Ā andĀ
.
Which is the greater quantity?
(a) TheĀ -coordinate of theĀ
-intercept of this line
(b) TheĀ -coordinate of theĀ
-intercept of this line
(a) is greater.
It is impossible to tell from the information given.
(a) and (b) are equal.
(b) is greater.
(a) is greater.
The slope of this line isĀ
.
We will use the point-slope form of the line, with this slope and pointĀ :
TheĀ -coordinate of theĀ
-intercept of this line can be found by substitutingĀ
Ā and solving forĀ
:
Ā
TheĀ -coordinate of theĀ
-intercept of this line can be found by substitutingĀ
Ā and solving forĀ
:
This makes (a) the greater quantity.
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All ISEE Upper Level Quantitative Resources
