# ISEE Upper Level Quantitative : How to multiply variables

## Example Questions

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### 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 : How To Multiply Variables

Multiply:

Explanation:

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

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

Simplify:

Explanation:

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

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

(a) is greater

Explanation:

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

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

N is a positive integer; . Which is greater?

(a)

(b)

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Explanation:

If , then .

If , then .

Therefore, at least two possibilities can be demonstrated.

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

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

Explanation:

Since ,

.

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

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 : How To Multiply Variables

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

(A)

(B)

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

(A) is greater

(B) is greater

(A) and (B) are equal

(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 #9 : How To Multiply Variables

is a positive number. Which is greater?

(A) One ninth of 117% of

(B) One eleventh of 143% of

(B) is greater

(A) and (B) are equal

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

(A) is greater

(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 #3 : How To Multiply Variables

Which of the following expressions is equivalent to

?