Stardust Chris wrote:

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.

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)

Our high-level "quant" preparation starts here: https://gmath.net