### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #151 : Equations

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

It can be deduced that both and are nonnegative, since both are radicands of square roots.

, so

, so

, and

.

### Example Question #151 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

**Correct answer:**

(b) is the greater quantity

By the Zero Product Principle, one of the factors is equal to 0:

which is impossible for any real value of , or

.

By the Zero Product Principle, one of the factors is equal to 0:

which is impossible for any real value of , or

Since and , it can be determined that .

### Example Question #152 : Equations

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

**Correct answer:**

(a) is the greater quantity

Between two fractions with the same numerator, the one with the lesser denominator is the greater, so

and .

### Example Question #151 : How To Find The Solution To An Equation

, , and all stand for positive quantities.

Which is the greater quantity?

**Possible Answers:**

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

Solve the equations for and in terms of :

Therefore, we seek to determine which of and is greater, bearing in mind that both of these quantities, as well as , must be positive.

We can make the following observation:

Suppose

Then

But if , then

and

, a contradiction.

Therefore, it must hold that , and .

### Example Question #153 : Equations

, , and all stand for positive quantities.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

**Correct answer:**

(a) is the greater quantity

Solve the equations for and in terms of :

and is positive, so by the properties of inequality,

### Example Question #154 : Equations

Solve for :

**Possible Answers:**

**Correct answer:**

### Example Question #155 : Equations

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

**Correct answer:**

(a) and (b) are equal

### Example Question #151 : How To Find The Solution To An Equation

Figure NOT drawn to scale

Above is a straight line on a graph. Which is the greater quantity?

(a)

(b) 18

**Possible Answers:**

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

**Correct answer:**

(a) is the greater quantity

If we go from the point (48, 60) to (24, 42), we see that if the first coordinate decreases by 24, the second decreases by 18. Going from (24, 42) to the point on the -axis, the first coordinate again decreases by 24, so the second coordinate again decreases by 18:

.

### Example Question #157 : Equations

The reciprocal of is between 2 and 4. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

**Correct answer:**

It is impossible to determine which is greater from the information given

, so

Also,

, so

Therefore, it possible for

,

,

or

,

making it inconclusive whether or is the greater.

### Example Question #160 : How To Find The Solution To An Equation

Solve for :

**Possible Answers:**

**Correct answer:**