# ISEE Upper Level Quantitative : How to find the solution to an equation

## Example Questions

### Example Question #51 : Algebraic Concepts

Define

Which is the greater quantity?

(a)

(b)

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) and (b) are equal

Explanation:

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

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

### Example Question #52 : Algebraic Concepts

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

.

Which is the greater quantity?

(a)

(b)

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(a) and (b) are equal

Explanation:

Convert  to Celsius by substituting :

and  are the same temperature.

### Example Question #53 : Algebraic Concepts

Define

is positive. Which is the greater quantity?

(a)

(b)

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

It is impossible to tell from the information given

Explanation:

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 : Algebraic Concepts

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: 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 Explanation: 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 : Algebraic Concepts

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.

(a) and (b) are equal

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) is greater

Explanation:

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 : Algebraic Concepts

Define an operation  on two real numbers as follows:

For every real :

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) is greater

Explanation:

(a)

(b)

### Example Question #57 : Algebraic Concepts

Quantity A: The number of days in  weeks.

Quantity B: The number of hours in  days.

Quantity A is greater.

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

The two quantities are equal.

Quantity B is greater.

Quantity A is greater.

Explanation:

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 #58 : 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?

Explanation:

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 #59 : Algebraic Concepts

Quantity A: The mean of .

Quantity B: The mean of .

Quantity A is greater.

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

Quantity B is greater.

The two quantities are equal.

Quantity B is greater.

Explanation:

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 : Algebraic Concepts

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

Cannot be determined

Cannot be determined

Explanation:

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.