### All ISEE Middle Level Quantitative Resources

## Example Questions

### Example Question #11 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Define an operation on the real numbers as follows:

For all real values of and ,

is a positive number. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is the greater quantity

(a) and (b) are equal

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

(a) is the greater quantity

**Correct answer:**

(a) and (b) are equal

so

and

The two are equal regardless of the value of .

### Example Question #12 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

and are both negative numbers. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

**Correct answer:**

(a) and (b) are equal

The two quantities are equal regardless of the values of and . To see this, we note that

and

Therefore, by the addition property of equality,

### Example Question #13 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which is the greater quantity?

(a)

(b) 18

**Possible Answers:**

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

**Correct answer:**

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

The information is insufficient, as we see by exploring two cases:

Case 1:

Case 2:

Remember, the three variables need not stand for *whole numbers.*

### Example Question #1 : How To Subtract Variables

What is the value of ?

**Possible Answers:**

**Correct answer:**

To solve for , the fractions should first be converted to ones that share a common denominator. Given that , the common denominator is 12.

Thus, can be converted to . This gives us:

### Example Question #2 : How To Subtract Variables

Simplify:

**Possible Answers:**

**Correct answer:**

It is easiest to begin by moving like terms together. Hence:

becomes

(Notice that is its own term.)

Now, consider the coefficients for each term.

For , you have

For , you have

Hence, the expression simplifies to:

This can be moved around to get the correct answer (which means the same thing):

### Example Question #2 : How To Subtract Variables

Simplify:

**Possible Answers:**

**Correct answer:**

Begin by distributing the two groups. Notice that you must distribute the subtraction through the groups:

becomes

Next, you should move like terms next to each other:

(Notice that is its own term.)

Now, combine terms.

For , you get

For , you get

Therefore, the final form of the expression is:

### Example Question #3 : How To Subtract Variables

Solve for :

**Possible Answers:**

**Correct answer:**

Begin by distributing. Thus,

becomes

(Don't forget that you have to distribute your subtraction for the second group.)

Combine like terms on the right side of the equation:

Next, move the values to the left side of the equation and all of the other values to the right side:

Combine like terms on the left:

Finally, divide everything by :

This comes out to be:

or

### Example Question #4 : How To Subtract Variables

Simplify:

**Possible Answers:**

**Correct answer:**

Begin by distributing the multiplied groups:

Next, move all similar factors together:

Now, combine each set of similar factors:

Therefore, our answer is:

### Example Question #5 : How To Subtract Variables

Simplify:

**Possible Answers:**

**Correct answer:**

This problem is not too difficult. Begin by moving all common terms next to each other:

Next, simplify each group of terms that has the same set of variables:

And do not forget that you are left with as well!

Now, combine all of these:

### Example Question #7 : How To Subtract Variables

Simplify:

**Possible Answers:**

**Correct answer:**

Begin by moving common factors next to each other. Thus,

becomes

Now, combine each set:

Remember, there still is also.

Therefore, the simplified form of the expression is: