Little correction: it should be Y + B = 13 and R + B = 7.

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?

We need to find Y - R.

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

This implies that Y + B = 13 because only if there are total of 13 blue and yellow marbles, removing 14 marbles guarantees getting a red one: you should pick all blue and yellow and the next one for sure will be red. But knowing that B + Y = 13 is not enough to get Y - R. Not sufficient.

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

The same ways as above this means that R + B = 7. Not sufficient.

(1)+(2) Subtract R + B = 7 from Y + B = 13: (Y + B) - (R + B) = 13 - 7 --> Y - R = 6. Sufficient.

Answer: C.
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\(? = Y - R\,\,\,\,\,\left( {{\text{yellow,}}\,{\text{red and}}\,\,{\text{blue}}\,\,{\text{marbles}}\,\,{\text{only}}} \right)\)

\(\left( 1 \right)\,\,Y + B = 13\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {Y,R,B} \right) = \left( {12,1,1} \right)\,\,\,\, \Rightarrow \,\,\,? = 11 \hfill \cr
\,{\rm{Take}}\,\,\left( {Y,R,B} \right) = \left( {12,2,1} \right)\,\,\,\, \Rightarrow \,\,\,? = 10 \hfill \cr} \right.\)

\(\left( 2 \right)\,\,R + B = 7\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {Y,R,B} \right) = \left( {12,6,1} \right)\,\,\,\, \Rightarrow \,\,\,? = 6 \hfill \cr
\,{\rm{Take}}\,\,\left( {Y,R,B} \right) = \left( {13,6,1} \right)\,\,\,\, \Rightarrow \,\,\,? = 7 \hfill \cr} \right.\)

\(\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
\,Y + B = 13 \hfill \cr
\,R + B = 7 \hfill \cr} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,\,? = Y - R = 13 - 7\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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