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

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Automobile A is traveling at two-thirds the speed that Automobile B  [#permalink]

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New post 17 Feb 2015, 10:46
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A
B
C
D
E

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Question Stats:

85% (01:22) correct 15% (01:45) wrong based on 154 sessions

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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
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Automobile A is traveling at two-thirds the speed that Automobile B  [#permalink]

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New post Updated on: 18 Dec 2018, 07:29
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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



Let‘s call Automobile A‘s current speed A and Automobile B‘s current speed B. The question stem tells us that \(A=\frac{2}{3}B\). Statement (1) tells us that \(A+10=\frac{3}{4}(B+10)\). Thus, between the question stem and Statement (1), we have two non-identical equations, which means we can solve for our two variables. Rule out B, C, and E.

Statement (2) tells us that \(A-10=\frac{1}{2}(B-10)\). Once again, we have two non-identical equations and two variables. Statement (2) is sufficient, and the correct answer is D.

Originally posted by Stardust Chris on 17 Feb 2015, 10:54.
Last edited by Bunuel on 18 Dec 2018, 07:29, edited 2 times in total.
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Re: Automobile A is traveling at two-thirds the speed that Automobile B  [#permalink]

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New post 17 Feb 2015, 10:57
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1
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



Speed of A = A
Speed of B = B

Then, A=(2/3)B

Question: 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.

(A+10) = (3/4) (B+10)
Substituting B from the previous relationship of A and B
(A+10) = (3/4) (3/2A+10)
8A + 80 = 9A + 60
i.e. A = 20
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

(A-10) = (1/2) (B-10)
Substituting B from the previous relationship of A and B
We can find the value of A
i.e. A = 20
SUFFICIENT

Answer: Option
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Re: Automobile A is traveling at two-thirds the speed that Automobile B  [#permalink]

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New post 18 Dec 2018, 07:10
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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


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

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Re: Automobile A is traveling at two-thirds the speed that Automobile B  [#permalink]

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New post 18 Dec 2018, 13:34
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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Re: Automobile A is traveling at two-thirds the speed that Automobile B   [#permalink] 18 Dec 2018, 13:34
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