### All ISEE Upper Level Quantitative Resources

## Example Questions

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

Simplify:

**Possible Answers:**

**Correct answer:**

Group and collect like terms:

### Example Question #31 : Variables

Simplify the expression:

**Possible Answers:**

The expression cannot be simplified further.

**Correct answer:**

The expression cannot be simplified further.

There are no like terms among these three; one has only as a variable, one has only , one has both. Therefore, no terms can be collected, and the expression cannot be simplified further.

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

Which is the greater quantity?

(a) The sum of the integers from 1 to 1,000

(b) The sum of the integers from 1,001 to 1,500

**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

(a) The sum of the integers from 1 to 1,000 is

(b) The sum of the integers from 1,001 to 1,500 is

(b) is greater

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

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(B) is greater

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

(A) and (B) are equal

(A) is greater

**Correct answer:**

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

Which quantity is the greater depends on the value of .

Case 1:

Then

and

Subsequently, (A) is greater

Case 2:

Then

and

Subsequently, (B) is greater

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

Evaluate for :

**Possible Answers:**

**Correct answer:**

Substitute and evaluate, remembering that in order of operations, squaring comes before adding:

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

Which of the following is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) is greater

(B) is greater

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

(A) and (B) are equal

**Correct answer:**

(B) is greater

(A) Substitute and evaluate, remembering that by order of operations, squaring precedes addition:

(B) Substitute and evaluate

(B) is greater

### Example Question #201 : Algebraic Concepts

is negative. Which of these quantities is the greater?

(A)

(B)

**Possible Answers:**

(A) is greater

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

(B) is greater

(A) and (B) are equal

**Correct answer:**

(B) is greater

Since and is negative, then by the multiplication property of inequality,

Also, , so by the addition property of inequality,

making (B) greater.

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

is positive. Which of these quantities is greater?

(A)

(B)

**Possible Answers:**

(B) is greater

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

(A) is greater

(A) and (B) are equal

**Correct answer:**

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

The values of both terms depend on the value of .

Case 1:

, making (B) greater.

Case 2:

, making (A) greater.

Therefore, the information given is insufficient.

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

When evaluating the expression

,

assuming you know the values of all five variables, what is the *third *operation that must be performed?

**Possible Answers:**

Subtraction

Addition

Multiplication

Division

Squaring

**Correct answer:**

Multiplication

The operation that must be performed first is the operation in parentheses, which is the division.

After this, the order of operations is:

Exponents

Multiplication and division (left to right)

Addition and subtraction (left to right)

The second thing that must be done is the exponentiation - the squaring. Of the three remaining operations, the multiplication takes precedence. The correct response is multiplication.

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

The Fibonacci sequence is defined as follows:

For integers , .

Which is the greater quantity?

(a) The sum of and

(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

Using the definition of the sequence for all terms beginning with the third, substitute 778 for :

The two quantities are equal.