### All ISEE Middle Level Quantitative Resources

## Example Questions

### Example Question #1 : How To Add Variables

Which quantity is greater if ?

**Possible Answers:**

is greater

is greater

and are equal

It is not possible to tell from the information given

**Correct answer:**

is greater

We know that is always positive for all values of . Therefore would be negative for all values of . From this conclusion, we know:

So we have:

is the greater quantity.

### Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater if ?

**Possible Answers:**

It is not possible to tell from the information given

is greater

and are equal

is greater

**Correct answer:**

It is not possible to tell from the information given

A positive number raised to the third power will be positive, while a negative number raised to the third power will remain negative.

If , then and .

If , then and .

Since we do not know if is positive or negative, we cannot draw a conclusion about which option is greater.

If , then is greater.

If , then is greater.

### Example Question #1 : Variables

Which quantity is greater if ?

**Possible Answers:**

is greater

It is not possible to tell from the information given

is greater

and are equal

**Correct answer:**

is greater

When we can write:

We know that and . Based on this, we can compare the two given quantities.

is the greater quantity.

### Example Question #1 : Variables

Which quantity is greater if ?

**Possible Answers:**

and are equal

is greater

It is not possible to tell from the information given

is greater

**Correct answer:**

is greater

We know that is greater than . We can easily test a few values for to determine if the values are increasing or decreasing.

If :

If :

If :

The value of is increasing, with the smallest possible value being . From this, we know that , so .

### Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which of the following is equivalent to ?

**Possible Answers:**

**Correct answer:**

Using the distributive property:

and

Using the associative property of multiplication:

We can rewrite as ; using the commutative and associative properties of multiplication:

is the sum of unlike terms and cannot be simplified.

is the correct choice.

### Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

is a positive integer.

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) is greater

(B) is greater

(A) and (B) are equal

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

Depending on the value of , it is possible for either expression to be greater or for both to be equal.

Case 1:

and

So the two are equal.

Case 2:

and

So (B) is greater.

The correct response is that it cannot be determined which is greater.

### Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

is a positive integer.

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) is greater

(A) and (B) are equal

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

(B) is greater

**Correct answer:**

(A) is greater

Since , and is positive,

then by the multiplication property of inequality,

making (A) greater regardless of the value of .

### Example Question #8 : Algebraic Concepts

is a positive integer.

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

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

(B) is greater

(A) is greater

(A) and (B) are equal

**Correct answer:**

(A) and (B) are equal

Regardless of the value of , the expressions are equal.

### Example Question #9 : Algebraic Concepts

Which of the following is equivalent to ?

**Possible Answers:**

None of the other responses is correct.

**Correct answer:**

None of the other responses is correct.

The expression is the sum of two unlike terms, and therefore cannot be further simplified. None of these responses is correct.

### Example Question #10 : Algebraic Concepts

is a positive integer.

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

Since , , so (A) is greater regardless of the value of .