### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #51 : Equations

Define

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) and (b) are equal

(a) To evaluate , use the definition for nonnegative values of :

(b) To evaluate , use the definition for negative values of :

### Example Question #51 : How To Find The Solution To An Equation

To convert Fahrenheit temperature to the equivalent in Celsius , use the formula

.

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

Convert to Celsius by substituting :

and are the same temperature.

### Example Question #53 : Equations

Define

is positive. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

**Correct answer:**

It is impossible to tell from the information given

Substitute for in the definition of :

The relationship of this expression to 0 is uncertain for positive , as we see with these two cases:

### Example Question #54 : Equations

Carl, the barista at Moose Jaw Coffee, has to mix together two kinds of coffee beans - Vanilla Heaven, which costs $10 a pound, and Mountain Goodness, which costs $15 a pound - to produce a coffee that costs $12 a pound. The beans in the mixture sell for the same price as they would separately.

Which is the greater quantity?

(a) The number of pounds of Vanilla Heaven beans he will use

(b) The number of pounds of Mountain Goodness beans he will use.

**Possible Answers:**

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

**Correct answer:**

(a) is greater

Let and be, respectively, the number of pounds of Vanilla Heaven and Mountain Goodness beans he will use. Then the cost of the Vanilla Heaven beans will be $10 a pound times the number of pounds, or ; similarly, the cost of the Mountain Goodness beans will be . There will be pounds of beans total selling for $12 a pound, so the total cost will be . Therefore, we have the equation:

We can solve for :

Two-thirds as much Mountain Goodness coffee goes into the mix as Vanilla Heaven.

### Example Question #55 : Equations

Consider the line of the equation .

Which is the greater quantity?

(a) The slope of this line

(b) The slope of a line perpendicular to this line.

**Possible Answers:**

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

**Correct answer:**

(a) is greater

The slope of the line can be found by rewriting it in slope-intercept form:

The slope of this line is the coeffcient of , which is , a positive number. Since the slope of a line perpendicular to this line is its opposite reciprocal, it must be negative. This makes (a) greater.

### Example Question #56 : Equations

Define an operation on two real numbers as follows:

For every real :

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

(a)

(b)

### Example Question #57 : Equations

Quantity A: The number of days in weeks.

Quantity B: The number of hours in days.

**Possible Answers:**

Quantity A is greater.

The two quantities are equal.

Quantity B is greater.

There is not enough information to determine the relationship of the quantities.

**Correct answer:**

Quantity A is greater.

Multiplying days per week by weeks, we find that there are days in weeks. Multiplying hours per day by days, we find that there are only hours in days.

### Example Question #51 : Algebraic Concepts

Jim has quarters and nickles that together total . If the number of quarters and nickles were switched, his total amount of money would be less. How many quarters does Jim have?

**Possible Answers:**

**Correct answer:**

If Jim has less when the number of coins is switched, then the switch scenario leaves him with .

The easiest way to solve this problem is to plug in the answer choices. If we plug in for the number of quarters he has, we find that he must also have nickles in order to equal . Reversing those numbers ( nickles and quarters) gives us the needed .

### Example Question #51 : How To Find The Solution To An Equation

Quantity A: The mean of .

Quantity B: The mean of .

**Possible Answers:**

The relationship of the quantities cannot be determined from the information provided.

Quantity A is greater.

The two quantities are equal.

Quantity B is greater.

**Correct answer:**

Quantity B is greater.

To find the mean, sum the terms and divide by the number of terms in the set.

Quantity A:

Quantity B:

Quantity B is larger.

### Example Question #60 : Equations

When is divided by , the remainder is . What is the remainder when is divided by ?

**Possible Answers:**

Cannot be determined

**Correct answer:**

Cannot be determined

Pick a number for that satisfies the condition for division by 15 and see what happens when it is divided by 7.

33 divided by 15 leaves a remainder of 3. 33 divided by 7 leaves a remainder of 5.

Let's try another number as well.

48 divided by 15 leaves a remainder of 3. 48 divided by 7 leaves a remainder of 6, which is different from the remainder left by 33.

Similarly, 63 divided by 15 leaves a remainder of 3. 63 divided by 7 leaves no remainder at all. Therefore, the answer cannot be determined.