ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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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 

Possible Answers:

(a) and (b) are equal

(a)Ā is greater

It is impossible to tell from the information given

(b) is greater

Correct answer:

(a) and (b) are equal

Explanation:

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.

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

(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 

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

It is impossible to tell from the information given.

Explanation:

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)

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

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)

Possible Answers:

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Correct answer:

(a) and (b) are equal

Explanation:

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

Possible Answers:

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

(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 

Possible Answers:

(a)Ā is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a)Ā is greater.

Explanation:

(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 

Possible Answers:

(a)Ā is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

(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)

Possible Answers:

It cannot be determined from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

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

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

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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