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

## Example Questions

### Example Question #41 : Algebraic Concepts

is defined as the greatest integer less than or equal to

is defined as the least integer greater than or equal to

itself is not an integer.

Which is the greater quantity?

(a)

(b)

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

Explanation:

Let  .

Then .

Since is a noninteger,  and .

(a)

(b)

(a) is greater than (b).

### Example Question #711 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Define  and .

can be any real number.

Which is the greater quantity?

(a)

(b)

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(b) is greater.

Explanation:

(a)

Substitute  for :

(b)

Substitute  for :

, so regardless of the value of , and (b) is greater.

### Example Question #43 : Algebraic Concepts

Veronica and Stephanie bought sweaters of the same type, but at different stores. Veronica bought hers for  at Markham's; Stephanie bought hers for  at Schuster's. Veronica got a better discount rate than Stephanie, and she has been bragging about it.

Which is the greater quantity?

(a) The price of the sweater at Markham's before discount

(b) The price of the sweater at Schuster's before discount

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

It is impossible to tell from the information given.

Explanation:

We show that the question cannot be answered with certainty by taking two sample cases.

Case 1: The sweaters cost  at both stores.

Veronica saved  and got a  discount rate. Stephanie saved  and got a discount rate.

Case 2: The sweater cost  at Markham's and  at Schuster's.

In this case Veronica saved  and got a  discount rate. Stephanie saved  and got a discount rate of .

### Example Question #44 : Algebraic Concepts

Define .

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Explanation:

(a)

(b)

### Example Question #45 : Algebraic Concepts

The distance between Vandalia and Clark is 250 miles and is represented by six inches on a map. The distance between Vandalia and Ferrell is 125 miles.

Which is the greater quantity?

(a) The distance between Clark and Ferrell on a map

(b) Two inches

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) is greater.

Explanation:

The actual distance between Clark and Ferrell is unknown, but it is at least  miles. If  is the map distance between Clark and Ferrell, then its minimum can be calculated using a proportion:

At the very least, the two are three inches apart on the map, so (a) is greater regardless.

### Example Question #46 : Algebraic Concepts

Carl starts working for a bicycle store today. He wants to buy a certain bicycle, but his  employee discount doesn't take effect until ninety days from now. He finds out that the price of the bicycle is due to increase by  in sixty days.

Which is the greater quantity?

(a) The price Carl would pay for the bicycle today

(b) The price Carl would pay for the bicycle in ninety days, assuming that the bicycle does not change in price again

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

(a) Let  be the current price of the bicycle. Without the discount and the markup, Carl would pay  today.

(b) Now we examine what Carl would pay in ninety days. The increase would be  of , or, equivalently, , and the price of the bicycle is

.

The employee discount will be  of this, or , and the price Carl would pay is

.

Carl would pay more by waiting ninety days.

### Example Question #47 : Algebraic Concepts

The distance between Vandalia and Clark is 250 miles and is represented by six inches on a map. The distance between Vandalia and Ferrell is 125 miles.

Which is the greater quantity?

(a) The distance between Clark and Ferrell on a map

(b) Ten inches

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

The actual distance between Clark and Ferrell is unknown, but at most it is  miles. If  is the map distance between Clark and Ferrell, then its maximum can be calculated using a proportion:

At the very most, the two are nine inches apart on the map, so (b) is greater regardless.

### Example Question #48 : Algebraic Concepts

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) and (b) are equal.

Explanation:

Substitute to evaluate each expression:

(a)

(b)

### Example Question #49 : Algebraic Concepts

A set of tires costs  after a  discount.

Which is the greater quantity?

(a) The price of the tires before the discount

(b)

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) is greater.

Explanation:

discount means that the tires sell for  of their original price. If that original price is , then after the discount, they will sell for  of . This is

,

which is less than . Therefore, the original purchase price must be greater than .

### Example Question #50 : Algebraic Concepts

Define .

Which is the greater quantity?

(a)

(b)

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:

(a)

First, evaluate :

,

so  .

Now, evaluate :

so .

(b)

First, evaluate :

,

so .

Now evaluate

,

so .