# ISEE Upper Level Quantitative : Variables

## Example Questions

### Example Question #931 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and  are positive integers. Which is the greater quantity?

(A)

(B)

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

(B) is greater

(A) is greater

(A) and (B) are equal

(B) is greater

Explanation:

Since  and  are positive,

for all positive  and , making (B) greater.

### Example Question #932 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and  are negative integers. Which is the greater quantity?

(A)

(B)

(A) and (B) are equal

(A) is greater

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

(B) is greater

(B) is greater

Explanation:

Since  and  are both negative,

.

for all negative  and , making (B) greater.

### Example Question #933 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and  are positive integers. Which is the greater quantity?

(A)

(B)

(A) is greater

(A) and (B) are equal

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

(B) is greater

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

Explanation:

It is impossible to tell which is greater.

Case 1:

Then

and

.

This makes (A) and (B) equal.

Case 2:

Then

and

.

This makes (A) the greater quantity.

### Example Question #91 : Variables

and  are positive integers greater than 1.

Which is the greater quantity?

(A)

(B)

(A) is greater

(A) and (B) are equal

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

(B) is greater

(A) is greater

Explanation:

One way to look at this problem is to substitute . Since  must be positive, and this problem is to compare  and

and

Since 2, , and  are positive, by closure, , and by the addition property of inequality,

Substituting back:

(A) is the greater quantity.

### Example Question #934 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and  are positive integers greater than 1.

Which is the greater quantity?

(A)

(B)

(A) and (B) are equal

(A) is greater

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

(B) is greater

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

Explanation:

Case 1:

Then

and

This makes the quantities equal.

Case 2:

Then

and

This makes (B) greater.

Therefore, it is not clear which quantity, if either, is greater.

### Example Question #92 : Variables

and  are positive integers greater than 1.

Which is the greater quantity?

(A)

(B)

(A) and (B) are equal

(B) is greater

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

(A) is greater

(A) is greater

Explanation:

One way to look at this problem is to substitute . The expressions to be compared are

and

Since  is positive, so is , and

Substituting back,

,

making (A) greater.

### Example Question #936 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Factor:

The expression is a prime polynomial.

Explanation:

We can rewrite as follows:

Each group can be factored - the first as the difference of squares, the second as a pair with a greatest common factor. This becomes

,

which, by distribution, becomes

### Example Question #93 : Variables

is a positive number;  is the additive inverse of .

Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

(b) is the greater quantity

Explanation:

If   is the additive inverse of , then, by definition,

.

, as the difference of the squares of two expressions, can be factored as follows:

Since , it follows that

Another consequence of  being the additive inverse of  is that

, so

is positive, so  is as well.

It follows that .

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

Half of one hundred divided by five and multiplied by one-tenth is __________.

5

1

10

2

1

Explanation:

Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.

Simplify: