swerve wrote:A garage has a stock of side mirrors. The ratio of right side mirrors to left side mirrors is 5:3. Iris, a garage worker, attaches pairs of left and right mirrors with an adhesive tape until no more pairs can be made. If 30 right mirrors are left unpaired, how many left and right mirrors are there in the stock?
A. 240
B. 120
C. 80
D. 75
E. 48
Source: Economist GMAT
Excellent opportunity to use the
k technique :
$$\left\{ \matrix{
{\rm{Right}} = 5k \hfill \cr
{\rm{Left}} = 3k \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\left( {k \ge 1\,\,{\mathop{\rm int}} \,\,\left( * \right)} \right)\,\,\,\,\,\,\,\,\,$$
$$\left( * \right)\,\,\,k = 2 \cdot \left( {3k} \right) - 5k = 2 \cdot {\mathop{\rm int}} - {\mathop{\rm int}} = {\mathop{\rm int}} $$
$$? = 8k\,\,\,\,\,\,\,\,\,\,\,$$
$$\left\{ \matrix{
\,\,3k\,\,{\rm{pairings}} \hfill \cr
\,\,30 = \left( {5k - 3k} \right) = 2k\,\,{\rm{Right}}\,{\rm{unpaired}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,2k = 30\,\,\,\,\,\,\,\mathop \Rightarrow \limits_{{\rm{FOCUS}}\,!}^{ \cdot \,\,4} \,\,\,\,\,\,\,? = 8k = 4 \cdot 30 = 120 \hfill \cr} \right.\,\,\,\,\,\,$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
