# ISEE Upper Level Quantitative : How to multiply variables

## Example Questions

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### Example Question #1 : Variables

Multiply:       Explanation:

This can be solved using the pattern for the sum times difference of two terms:    ### Example Question #1 : How To Multiply Variables

Multiply:       Explanation:

This can be solved using the pattern for the sum times difference of two terms:    ### Example Question #2 : Operations

Simplify:       Explanation:

This can be solved using the pattern for the square of a binomial:    ### Example Question #2 : Operations

N is a positive integer; . Which is greater?

(a) (b) 0

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

(a) is greater

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

### Example Question #1 : Variables

N is a positive integer; . Which is greater?

(a) (b) (b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

It is impossible to tell from the information given.

Explanation:

If , then .

If , then .

Therefore, at least two possibilities can be demonstrated.

### Example Question #1 : Variables

Which is the greater quantity?

(a) (b) It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:    Since , .

### Example Question #3 : Operations

Assume is nonzero. Which expression is not equivalent to ?   All four expressions given in the other choices are equivalent to .  Explanation:

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 : Variables

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

(A) (B) (A) is greater

(A) and (B) are equal

(B) is greater

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

(B) is greater

Explanation: ,

So by the multiplication property of inequality, when each is multiplied by the negative number , and (B) is greater.

### Example Question #4 : Operations is a positive number. Which is greater?

(A) One ninth of 117% of (B) One eleventh of 143% of It is impossible to determine which is greater from the information given

(A) is greater

(B) is greater

(A) and (B) are equal

(A) and (B) are equal

Explanation:

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 #1 : How To Multiply Variables

Which of the following expressions is equivalent to ?      Explanation:

Use the difference of squares pattern as follows:    ← Previous 1 3

### All ISEE Upper Level Quantitative Resources 