Automobile A is traveling at two-thirds the speed that Automobile B

Given: Automobile A is traveling at two-thirds the speed that Automobile B is traveling.
Let A = Car A's speed
Let B = Car B's speed
So, we can write: A = (2/3)B

Target question: What is the value of A?

Statement 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.
Car A's new speed = A + 10
Car B's new speed = B + 10
So, we can write: A + 10 = (3/4)(B + 10)
We already know that: A = (2/3)B
IMPORTANT: Since we have a system of 2 different linear equations with 2 variables, we COULD solve the system for A and B (but we'd never waste valuable time on test day doing so)
So, we COULD answer the target question with certainty.
Statement 1 is SUFFICIENT

Statement 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
Car A's new speed = A - 10
Car B's new speed = B - 10
So, we can write: A - 10 = (1/2)(B - 10)
We already know that: A = (2/3)B
Once again, we have a system of 2 different linear equations with 2 variables, which we COULD solve for A and B.
Statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

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.

its a very tricky..question.

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