ISEE Upper Level Quantitative : How to find the solution to an equation

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

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Example Questions

Example Question #31 : Equations

Which is the greater quantity?

(a) 

(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:

(a) and (b) are equal

Explanation:

Multiply both sides of the top equation by 3, and subtract both sides of the second equation.

          

 

Now substitute to find :

 

The two are equal.

Example Question #32 : Equations

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Triple both sides of the top equation, and subtract both sides of the bottom equation:

  

           

Now substitute to find :

This makes 

Example Question #33 : Equations

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

We show that it cannot be determined which of  and , if either, is greater, by showing one case in which  and one case in which .

Case 1: . Then 

and .

Case 2: . Then 

and .

Example Question #34 : Equations

Which is the greater quantity?

(a) 

(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) is greater

Explanation:

If , then, since  is nonnegative, .

If , then , so the equation becomes . Therefore, .

Either way, (a) is greater.

Example Question #711 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which is the greater quantity?

(a) 

(b) 50 

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

By the commutative property, 

 

By the multiplication property of equality,

By distribution,

Example Question #712 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

 is a positive number. Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is greater 

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Correct answer:

(b) is greater 

Explanation:

If , then 

If , then 

If , then 

Therefore, regardless of the value of , as long as  is positive, , and (b) is greater.

Example Question #713 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which is the greater quantity?

(a) 

(b)

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

From the multiplication property of equality,

.

By distribution,

.

Example Question #714 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

Take the absolute value of both sides:

This makes (a) greater.

 

Example Question #715 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

(a) 

(b) 

Rewrite as a fraction:

Example Question #716 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Define  as follows:

Which is the greater quantity?

(a) 

(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:

(b) is greater.

Explanation:

(a)  can be evaluated by using the definition of  for positive :

 can be evaluated by using the definition of  for nonpositive :

Add: 

 

(b)  can be evaluated by using the definition of  for nonpositive :

 

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