### All ISEE Upper Level Quantitative Resources

## Example Questions

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

Multiply:

**Possible Answers:**

**Correct answer:**

This can be solved using the pattern for the sum times difference of two terms:

### Example Question #2 : How To Multiply Variables

Multiply:

**Possible Answers:**

**Correct answer:**

This can be solved using the pattern for the sum times difference of two terms:

### Example Question #3 : How To Multiply Variables

Simplify:

**Possible Answers:**

**Correct answer:**

This can be solved using the pattern for the square of a binomial:

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

N is a positive integer; . Which is greater?

(a)

(b) 0

**Possible Answers:**

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

**Correct answer:**

(a) is greater

. Since , , and . This makes (a) greater.

### Example Question #5 : How To Multiply Variables

N is a positive integer; . Which is greater?

(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:**

It is impossible to tell from the information given.

If , then .

If , then .

Therefore, at least two possibilities can be demonstrated.

### Example Question #6 : How To Multiply Variables

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) is greater.

Since ,

.

### Example Question #7 : How To Multiply Variables

Assume is nonzero. Which expression is not equivalent to ?

**Possible Answers:**

All four expressions given in the other choices are equivalent to .

**Correct answer:**

Each expression can be simplified using the properties of exponents, among others:

The correct choice is , as it is the only choice not equivalent to .

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

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

(A)

(B)

**Possible Answers:**

(B) is greater

(A) is greater

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

(A) and (B) are equal

**Correct answer:**

(B) is greater

,

So by the multiplication property of inequality, when each is multiplied by the negative number ,

and (B) is greater.

### Example Question #9 : How To Multiply Variables

is a positive number. Which is greater?

(A) One ninth of 117% of

(B) One eleventh of 143% of

**Possible Answers:**

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

(A) is greater

(A) and (B) are equal

(B) is greater

**Correct answer:**

(A) and (B) are equal

117% of is equal to .

One-ninth of this is

143% of is equal to

One-eleventh of this is

Regardless of the value of , the quanitites are equal.

### Example Question #10 : How To Multiply Variables

Which of the following expressions is equivalent to

?

**Possible Answers:**

**Correct answer:**

Use the difference of squares pattern as follows: