AAPL wrote:Princeton Review

Automobile A is traveling at two-thirds the speed that Automobile B is traveling. How fast is Automobile A traveling?

1) If both automobiles increased their speed by 10 miles per hour, Automobile A would be traveling at three-quarters the speed that Automobile B would be traveling.

2) If both automobiles decreased their speed by 10 miles per hour, Automobile A would be traveling at half the speed that Automobile B would be traveling.

Excellent opportunity for the k technique, one of our powerful tools when dealing with ratios!
$$A = {2 \over 3}B\,\,\,\,\,\mathop \Rightarrow \limits^{B\,\, \ne \,\,0} \,\,\,\,\,{A \over B} = {2 \over 3}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,A = 2k \hfill \cr
\,B = 3k \hfill \cr} \right.\,\,\,\,\,\left( {k > 0} \right)\,\,\,\,\,\,\,\,\,\left[ {\,k\,\,{\rm{in}}\,\,{\rm{mph}}\,} \right]$$
$$? = A\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{\,? = k\,}$$

$$\left( 1 \right)\,\,\,2k + 10 = {3 \over 4}\left( {3k + 10} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,k\,\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,\,2k - 10 = {1 \over 2}\left( {3k - 10} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,k\,\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.