Today, we present the explanations for questions #5-6 from our Tricky Data Sufficiency Questions challenge.

Be sure to check out the explanations for questions #1-2, and questions #3-4 before moving on to today’s post. Let’s get started.

## Question #5

5. What is the value of *m*+*n*?

- (1)

*mn*= -8

(2) -2

*m*=

*n*

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient

There’s no obviously useful way to rephrase the question, so we’ll probably need to solve for *m* and *n *and then add the values together. Keep an eye open, though, for an opportunity to solve for the sum *m + n* directly.

Statement (1) doesn’t allow any good method of breaking up the single term *mn*, so will not yield a value for *m*, *n*, or *m + n*. Eliminate A and D.

Was that too fast? Suppose that it wasn’t obvious to you that the statement “doesn’t allow any good method of breaking up the single term *mn*.” What could you do? You could either *attempt to isolate the expression for which you’re solving* or *test values*. Let’s try the latter.

To test values for Statement (1) make a simple table. Use this table to ask, “Does every pair of values legal for Statement (1) yield the same answer to our question, that is the same value for *m + n?” *My table begins with a pair of values *m* and *n* whose product is -8, and adds a third column for the sum of those values. If the sum is the same for every row, then the statement is sufficient to answer the question, or we’ve just overlooked a counterexample.

Since different values consistent with Statement (1) yield different answers to our question, (1) is not sufficient. Eliminate A and D.

We could use a similar method with Statement (2), but it’s probably easier in this case to *isolate the expression for which we’re solving*.

-2m = n

Add *m* to each side to get *m*+*n* on the right-hand side of the equation.

-m = m + n

Since the expression *m + n *is not equal to a unique constant, unless we’ve missed some more fruitful way to manipulate this equation, Statement (2) doesn’t answer the question. Eliminate B.

Now consider the two statements together. Since Statement (1) is not linear, even together the two statements probably won’t yield a unique constant solution. There are a couple of exceptional cases though, so let’s solve this system of equations.

Since (2) gives us *n* in terms of *m*, it makes solving by substitution pretty manageable. Substitute -2*m* for *n* in (1).

m(-2m) = -8

Clear the parentheses.

Divide each side by -2.

Take the square root of each side of the equation.

That’s two possible solutions, so even together (1) and (2) are not sufficient to solve for *m*. It might occur to you that we’re after the sum *m + n *not just after m. If you want to be extra-careful, substitute the two possible values for m into either of the given equations and see whether they yield the same value for *m + n. *They do not. The correct answer is E.

## Question #6

6. If *xy* = 12, what is the value of *x* – *y*?

- (1)

*x*>

*y*

(2)

*x*+

*y*= 7

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient

In this case, the question includes an “if” statement, xy = 12.

Once again there is no obviously useful way to rephrase the question. We could, say, use the “if” equation to solve for *y* in terms of *x*, and substitute the resulting expression for *y* in the question. That’s not vey promising, though, so let’s turn to (1).

Even if we restrict ourselves to positive integers, the “if” statement and (1) together allow a number of solutions, each of which yields a different value for x – y. Specifically, *x *and *y* could be 12 and 1, 6 and 2, or 4 and 3, and could therefore be 11, 4, or 1. Eliminate A and D.

Statement (2) and the “if” statement together might seem to allow just one solution, *x*=4, *y*=3, and *x*–*y*=1. In fact, though, (2) and the “if” statement also allow *x*=3, *y*=4, =-1. So (2) is also not sufficient. Eliminate B.

Together (1) and (2) (along with the “if” statement) allow just one solution, *x*=4, *y*=3, and *x*–*y*=1. The correct answer is C.

**Remember to check back tomorrow for the explanations to questions #7 and #8. In the meantime, you can check your answers here.**

For question 5, isn’t a valid solution for the 2 equations: m = 2 and n = -4 such that the answer would be C?