### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #21 : Algebraic Concepts

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.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

**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 #22 : How To Find The Solution To An Equation

refers to the greatest integer less than or equal to .

and are integers.

Which is greater?

(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:**

(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 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater.

(a) is greater.

(a) and (b) are equal.

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 #24 : How To Find The Solution To An Equation

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.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

**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 #21 : How To Find The Solution To An Equation

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.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**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 #21 : Equations

.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(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 #21 : How To Find The Solution To An Equation

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:**

(b) is greater.

(a) and (b) are equal.

(a) 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 #22 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

**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 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**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 #30 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

**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,