ISEE Upper Level Quantitative : How to add variables

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

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Example Question #11 : How To Add Variables

 is a negative integer. Which is the greater quantity?

(A)

(B)

Possible Answers:

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

 

 

, so

and

is the greater quantity, regardless of .

Example Question #42 : Variables

 is a positive number.

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

Correct answer:

It cannot be determined which of (a) and (b) is greater

Explanation:

By examining two scenarios, we see that we cannot determine which is the greater quantity.

 

Case 1: 

Then 

and 

This makes (a) the greater quantity.

 

Case 2: 

Then 

and 

This makes (b) the greater quantity.

Example Question #42 : Operations

 is a negative number.

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

The expressions  and  are each other's opposite, so they have the same absolute value. Therefore, regardless of the value of 

.

Example Question #11 : How To Add Variables

 is a negative integer. Which is the greater quantity?

(A)

(B)

Possible Answers:

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

,

so

and

 regardless of the value of .

 

 

 

Example Question #45 : Variables

 is a negative integer. Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is the greater quantity

(b) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

Since

,

it follows that

and

regardless of the value of .

Example Question #51 : Variables

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

Suppose  is nonnegative.

Then 

Consequently,

,

which must be positive,

and 

,

which is the opposite of  and consequently must be negative. Therefore, (a) is greater.

 

Suppose  is negative. 

Then .

Consequently,

,

and 

.

, so

,

and (a) is greater.

 

(a) is the greater quantity either way.

 

Example Question #52 : Variables

Define . The graph of  is a line with slope .

.

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is the greater quantity

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

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

, so .

 

, so, by definition, , or .

The graph of  is a line through the point with coordinates  and with slope . The equation of the line can be determined by setting  in the slope-intercept form:

.

The equation of the line is , which makes this the definition of . By setting ,

.

Therefore, 

Example Question #51 : Variables

 and  are both positive.

Which is the greater quantity?

(a) 

(b) 

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

Explanation:

If ,

then 

.

The absolute value of a negative number is its (positive) opposite, so

Also, if  and  are both positive, then  is positive; the absolute value of a positive number is the number itself, so . Since , it follows that . Therefore, 

Since  is given to be positive,

and

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