### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #931 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are positive integers. Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(B) is greater

(A) is greater

It is impossible to tell which is greater from the information given

(A) and (B) are equal

**Correct answer:**

(B) is greater

Since and are positive,

for all positive and , making (B) greater.

### Example Question #932 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are negative integers. Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(B) is greater

(A) is greater

(A) and (B) are equal

It is impossible to tell which is greater from the information given

**Correct answer:**

(B) is greater

Since and are both negative,

.

for all negative and , making (B) greater.

### Example Question #933 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are positive integers. Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) and (B) are equal

(A) is greater

(B) is greater

It is impossible to tell which is greater from the information given

**Correct answer:**

It is impossible to tell which is greater from the information given

It is impossible to tell which is greater.

Case 1:

Then

and

.

This makes (A) and (B) equal.

Case 2:

Then

and

.

This makes (A) the greater quantity.

### Example Question #11 : How To Multiply Exponential Variables

and are positive integers greater than 1.

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) is greater

(A) and (B) are equal

(B) is greater

It is impossible to determine which is greater from the information given

**Correct answer:**

(A) is greater

One way to look at this problem is to substitute . Since , must be positive, and this problem is to compare and .

and

Since 2, , and are positive, by closure, , and by the addition property of inequality,

Substituting back:

(A) is the greater quantity.

### Example Question #934 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are positive integers greater than 1.

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) is greater

(A) and (B) are equal

(B) is greater

It is impossible to determine which is greater from the information given

**Correct answer:**

It is impossible to determine which is greater from the information given

Case 1:

Then

and

This makes the quantities equal.

Case 2:

Then

and

This makes (B) greater.

Therefore, it is not clear which quantity, if either, is greater.

### Example Question #932 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are positive integers greater than 1.

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

It is impossible to determine which is greater from the information given

(A) is greater

(B) is greater

(A) and (B) are equal

**Correct answer:**

(A) is greater

One way to look at this problem is to substitute . The expressions to be compared are

and

Since is positive, so is , and

Substituting back,

,

making (A) greater.

### Example Question #937 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Factor:

**Possible Answers:**

The expression is a prime polynomial.

**Correct answer:**

We can rewrite as follows:

Each group can be factored - the first as the difference of squares, the second as a pair with a greatest common factor. This becomes

,

which, by distribution, becomes

### Example Question #938 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

is a positive number; is the additive inverse of .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

**Correct answer:**

(b) is the greater quantity

If is the additive inverse of , then, by definition,

.

, as the difference of the squares of two expressions, can be factored as follows:

Since , it follows that

Another consequence of being the additive inverse of is that

, so

is positive, so is as well.

It follows that .