### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #21 : Equations

refers to the least integer greater than or equal to .

and are integers.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) is greater.

(a) Since is an integer, .

Since is an integer, .

(b) By closure, is an integer, so

.

(a) is the greater quantity.

### Example Question #21 : Algebraic Concepts

refers to the greatest integer less than or equal to .

and are integers.

Which is greater?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

**Correct answer:**

(b) is greater.

(a) Since is an integer, .

Since is an integer, .

(b) By closure, is an integer, so

.

This makes (b) greater.

### Example Question #23 : Equations

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It cannot be determined from the information given.

**Correct answer:**

(a) and (b) are equal.

Substitute and, subsequently, :

Factor as , replacing the two question marks with integers whose product is and whose sum is . These integers are .

Break this up into two equations, replacing for :

or

This has no solution, since must be nonnegative.

is the only solution, so (a) and (b) must be equal.

### Example Question #22 : Algebraic Concepts

Consider the line through points and .

Which is the greater quantity?

(a) The -coordinate of the -intercept of this line

(b) The -coordinate of the -intercept of this line

**Possible Answers:**

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

**Correct answer:**

(a) is greater.

The slope of this line is

.

We will use the point-slope form of the line, with this slope and point :

The -coordinate of the -intercept of this line can be found by substituting and solving for :

The -coordinate of the -intercept of this line can be found by substituting and solving for :

This makes (a) the greater quantity.

### Example Question #25 : Equations

The slope of a line is 2; the line does not pass through the origin.

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -intercept

**Possible Answers:**

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

**Correct answer:**

It is impossible to tell from the information given.

Let be the - and -intercepts, respectively. We know that the line does not pass through the origin - so .

Then the slope is:

Either or can be the greater. For example, if , then , and if , then .

### Example Question #26 : Equations

.

Which is the greater quantity?

(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:**

(a) and (b) are equal.

, so substitute and use the power of a power rule.

This makes (a) and (b) equal.

### Example Question #27 : Equations

Which is the greater quantity?

(a) The slope of the line of the equation

(b) The slope of the line of the equation

**Possible Answers:**

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) is greater.

Both equations are in slope-intercept form, so compare the coefficients of . The coefficients in (a) and (b) are 5 and 4, respectively, so these are the slopes of the lines. The line in (a) has the greater slope.

### Example Question #23 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

It is impossible to tell from the information given.

Using two different cases, we show that it is impossible to tell which is greater.

Case 1: . Then , and .

Case 2: . Then , and .

### Example Question #29 : Equations

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

**Correct answer:**

(a) is greater.

To solve the system of equations, add the left and right sides of the equation separately:

Divide:

Substitute to get :

is greater.

### Example Question #24 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

**Correct answer:**

It is impossible to tell from the information given

We show that it is possible for either or to be the greater by giving one of each case.

Case 1: . Then , so

Case 2: . Then , so

In Case 1, ; in Case 2,

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