Question

Of the four numbers represented on the number line above, is r closest to zero?
(1) q = -s
(2) -t < q

• 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.

Answer 1

The correct option is A Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

Referring to the figure above, in which it may be assumed that q, r, s, and t are different numbers, determine if r is closest to 0.

1. Since q = -s one of q and s is positive and the other is negative. Since s is to the right of q, then s is positive and q is negative. Also, 0 is halfway between q and s, so q and s are the same distance from 0. If r = 0 then, then, of q, r, s, and t, r is closest to 0 because it IS 0. If r is not equal to 0, then either (i) q < 0 < r < s < t or (ii) q < r < 0 < s < t.

(i) If q < 0 < r < s < t, as shown in hint figure (2) r is closer to 0 than s is because r is between 0 and s, and r is clearly closer to 0 than t is because t is farther away from 0 than s is. Also, since q and s are the same distance from 0 and r is closer to 0 than s is, then r is closer to 0 than q is. Therefore, r is closest to 0.

(ii) If q < r < 0 < s < t, as shown in hint figure (1), r is closer to 0 than q is because r is between 0 and q. Also, r is closer to 0 than s is because r is closer to 0 than q is and q and s are the same distance from 0. Moreover, r is closer to 0 than t is because t is farther away from 0 than s is. Therefore, r is closest to 0. In each case, r is closest to 0; SUFFICIENT.

1. If , -t < q , then −t is to the left of q. If t = 5, s = 4, r = 3 and q = -2, then -5 < -2 , so (2) is satisfied. In this case, q is closest to 0. On the other hand, if t = 5, s = 4, r = -1 and q = -2, then -5 < -2, so (2) is satisfied, but r is closest to 0; NOT sufficient.

• The correct answer is A; statement 1 alone is sufficient.