### All ISEE Upper Level Quantitative Resources

## Example Questions

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

Simplify:

**Possible Answers:**

**Correct answer:**

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

Simplify:

**Possible Answers:**

**Correct answer:**

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

is negative. Which of these quantities is the greater?

(A)

(B)

**Possible Answers:**

(A) is greater

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

(B) is greater

(A) and (B) are equal

**Correct answer:**

(B) is greater

,

So by the multiplication property of inequality, when each is multiplied by the negative number ,

.

Also,

,

so by the addition property of inequality,

or

This makes (B) greater.

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

Assume you know the values of all four variables in the expression

In which order do you perform the operations in order to evaluate the expression?

**Possible Answers:**

Multiply, divide, subtract

Subtract, divide, multiply

Subtract, multiply, divide

Multiply, divide, subtract

Divide, multiply, subtract

**Correct answer:**

Divide, multiply, subtract

Multiplication and division take precedence over subtraction in the order of operations, so these two operations are performed first. The two must be performed from left to right, so the division is worked first, followed by the multiplication. The subtraction is last.

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

Consider the expression:

Which of the following expressions *must* be equal in value to the above expression?

I)

II)

III)

**Possible Answers:**

I and II only

III only

I and III only

I, II, and III

I only

**Correct answer:**

I and III only

The order of operations is as follows:

Exponents

Multiplication and division (left to right)

Addition and subtraction (left to right)

The expression

is therefore evaluated by multiplying, then dividing, then adding. The net result is that the product is added to the quotient .

If we examine (I), we see that, since the multiplication is in parentheses, it is worked first. The division is worked second, then the addition. The order of operations has not changed, so the expressions are equivalent.

If we examine (II), we see that the order of operations has changed so that the addition is worked first. We see through example that the expressions can have different values:

If we examine (III), we see that, since the division is in parentheses, it is worked first. The multiplication is worked second, then the addition. The upshot is the same as in the main expression, however - the product is added to the quotient . Therefore, the expressions are equivalent.

The correct response is (I) and (III)

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

When evaluating the expression

,

assuming you know the values of all five variables, what is the *third *operation that must be performed?

**Possible Answers:**

The rightmost addition

The leftmost subtraction

The rightmost subtraction

The leftmost addition

The middle subtraction

**Correct answer:**

The leftmost subtraction

In the order of operations, any operations in parentheses must be performed first - there are two, the leftmost addition and the middle subtraction. What remains are the leftmost subtraction, the rightmost subtraction, and the rightmost addition. Since additions and subtractions are performed from left to right, the next, or third, operation performed is the leftmost subtraction.

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