Since the quotient of negative numbers is positive, both results will be positive.

We can rewrite both of these as quotients of positive numbers, as follows:

Since the expressions have the same dividend and the second has the greater divisor, the first has the greater quotient.

Therefore, (A) is greater.

The quotient of two negative numbers is positive. The expressions can be rewritten as follows:

Both expressions have the same dividend; the second has the lesser divisor so it has the greater quotient. This makes (B) greater.

In the order of operations, any expression within parentheses must be performed first. Between the parentheses, there are two operations, an exponentiation (squaring), and a subtraction. By the order of operations, the exponentiation is performed first; the subtraction is performed second, making this the correct response.

Since  is negative, its reciprocal  is also negative. Since 

,

by the Multiplication Property of Inequality,

That is, the reciprocal of  is greater than that of .

We show that the given information is insufficient by examining two cases.

Case 1: 

The reciprocal of  is , or .

Also, , the reciprocal of which is .

, so (b) is the greater quantity.

Case 2: .

The reciprocal of  is , or 2.

Also, , the reciprocal of which is .

, so (a) is the greater quantity.

 in both cases, but in one case, (a) is greater and in the other, (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.

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.

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)