Data Sufficiency

A certain number of marbles are to be removed from a box containing only solid-colored red, yellow, and blue marbles. How many more yellow marbles than red marbles are in the box before any are removed?

(1) To guarantee that a red marble is removed, the smallest number of marbles that must be removed from the box is 14.

(2) To guarantee that a yellow marble is removed, the smallest number of marbles that must be removed from the box is 8.

Plz help me understand this question. Thanks.

A certain number of marbles are to be removed from a box containing only solid-colored red, yellow, and blue marbles. How many more yellow marbles than red marbles are in the box before any are removed?

(1) To guarantee that a red marble is removed, the smallest number of marbles that must be removed from the box is 14.

(2) To guarantee that a yellow marble is removed, the smallest number of marbles that must be removed from the box is 8.

Question stem: What is the value of Y-R?

Statement 1: To guarantee that a red marble is removed, the smallest number of marbles that must be removed from the box is 14.
Implication:
It is possible to remove 13 NON-RED marbles, with the result that AT LEAST 14 marbles must be removed to guarantee that a red marble is removed.
Since there are 13 non-red marbles, we get:
Y+B = 13.
No way to determine the value of Y-R.
INSUFFICIENT.

Statement 2: To guarantee that a yellow marble is removed, the smallest number of marbles that must be removed from the box is 8.
Implication:
It is possible to remove 7 NON-YELLOW marbles, with the result that AT LEAST 8 marbles must be removed to guarantee that a yellow marble is removed.
Since there are 7 non-yellow marbles, we get:
B+R = 7.
No way to determine the value of Y-R.
INSUFFICIENT.

Statements combined:
Subtracting B+R=7 from Y+B=13, we get:
(Y+B) - (B+R) = 13-7
Y-R = 6.
SUFFICIENT.

The correct answer is C.

A certain number of marbles are to be removed from a box containing only solid-colored red, yellow, and blue marbles. How many more yellow marbles than red marbles are in the box before any are removed?

(1) To guarantee that a red marble is removed, the smallest number of marbles that must be removed from the box is 14.

(2) To guarantee that a yellow marble is removed, the smallest number of marbles that must be removed from the box is 8.

Let's say we have 'r' red marbles, 'y' yellow marbles, and 'b' blue marbles. If we want to know how many more yellow than red marbles we have, we're looking for y - r.

S1: If we need to select 14 marbles to guarantee that we remove a red, that means that there must be a total 13 blue and yellow marbles combined. (After we removed all 13 blue and yellow, the 14th pick would have to be red.) So this tell us that b + y = 13.

Well, that doesn't help much. You could say we have 1 blue and 12 yellow marbles, or 2 blue and 11 yellow marbles,etc. But we know nothing about how many red marbles there are. (Though presumably, there are fewer red than yellow.) Not Sufficient.


S2: If we need to select 8 marbles to guarantee that we remove a yellow, that means that there must be a total 7 blue and red marbles combined. (After we removed all 7 blue and red, the 8th pick would have to be yellow.) So this tell us that b + r = 7.

Also, not so helpful. You could say we have 1 blue and 6 red, or 2 blue and 5 red, etc. And we know nothing about how many yellow marbles there are.

Together: We know b + y = 13 and b + r = 7.

Subtract the second equation from the first:

b + y = 13.
-(b + r) = 7

And we get: y - r = 6. This is what we're looking for, so together, they are sufficient. There are 6 more yellow than red marbles. Answer is C