### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #21 : Variables

Evaluate .

**Possible Answers:**

**Correct answer:**

is the product of the sum and the difference of the same two binomials, so this can be rewritten, and evaluated, using the difference of squares pattern:

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

The ratio of 10 to 14 is closest to what value?

**Possible Answers:**

0.71

0.57

0.04

0.24

**Correct answer:**

0.71

Another way to express ratios is through division. 10 divided by 14 is approximate 0.71.

### Example Question #21 : Operations

If * *is the quotient of * *and , which statement could be true?

**Possible Answers:**

**Correct answer:**

A quotient is the result of division. If is the quotient of and , that means that could be true.

### Example Question #22 : Operations

is a negative integer. 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:**

(B) is greater

Since the quotient of negative numbers is positive, both results will be positive.

We can rewrite both of these as products of positive numbers, as follows:

, so

, and

making (B) greater.

### Example Question #4 : How To Divide Variables

is a negative integer. 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

Since the quotient of negative numbers is positive, both results will be positive.

We can rewrite both of these as quotients of positive numbers, as follows:

Since the expressions have the same dividend and the second has the greater divisor, the first has the greater quotient.

Therefore, (A) is greater.

### Example Question #22 : Variables

Let be negative. Which of the following is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) and (B) are equal

(B) is greater

(A) is greater

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

**Correct answer:**

(B) is greater

The quotient of two negative numbers is positive. The expressions can be rewritten as follows:

Both expressions have the same dividend; the second has the lesser divisor so it has the greater quotient. This makes (B) greater.

### Example Question #23 : Operations

When evaluating the expression

,

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

**Possible Answers:**

Multiplication

Squaring

Subtraction

Division

Addition

**Correct answer:**

Subtraction

In the order of operations, any expression within parentheses must be performed first. Between the parentheses, there are two operations, an exponentiation (squaring), and a subtraction. By the order of operations, the exponentiation is performed first; the subtraction is performed second, making this the correct response.

### Example Question #24 : Operations

is a negative number.

Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

**Possible Answers:**

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

Since is negative, its reciprocal is also negative. Since

,

by the Multiplication Property of Inequality,

That is, the reciprocal of is greater than that of .

### Example Question #8 : How To Divide Variables

; .

Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

**Possible Answers:**

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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

We show that the given information is insufficient by examining two cases.

Case 1:

The reciprocal of is , or .

Also, , the reciprocal of which is .

, so (b) is the greater quantity.

Case 2: .

The reciprocal of is , or 2.

Also, , the reciprocal of which is .

, so (a) is the greater quantity.

in both cases, but in one case, (a) is greater and in the other, (b) is greater.

### Example Question #25 : Variables

Simplify:

**Possible Answers:**

**Correct answer:**