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Two cars travel on a highway in the same direction. If car A travels

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Two cars travel on a highway in the same direction. If car A travels  [#permalink]

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New post Updated on: 07 Sep 2018, 21:23
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Two cars travel on a highway in the same direction. If car A travels at a rate of 75 mph and is \(y\) miles ahead of car B, which is traveling at 60 mph, in terms of \(y\), how many miles must car A travel to double the distance between itself and car B?


A) 3y
B) 4y
C) 5y
D) 6y
E) 8y

Originally posted by jennysussna on 07 Sep 2018, 13:32.
Last edited by generis on 07 Sep 2018, 21:23, edited 2 times in total.
Edited and formatted the question, renamed the topic
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Re: Two cars travel on a highway in the same direction. If car A travels  [#permalink]

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New post 07 Sep 2018, 15:08
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jennysussna wrote:
two cars are traveling on a highway in the same direction. if car A is traveling at a rate of 75 mph and is y miles ahead of car b, which is traveling at 60mpg, in terms of y, how many miles must car A travel to double the distance between itself and car b?

a 3y
b 4y
c 5y
d 6y
e 8y


This is an excellent opportunity to use "relative velocity" (in what we call the chasing scenario), carefully explained in our method!

The "starting point" is shown in the figure attached.

Let B stay still (virtual velocity zero) and A travel at a virtual speed of 15mph (75-60).

The temporary focus is (always!) the virtual time (equal to the real one!), in this case, time for A to travel y (extra) miles in its virtual speed....

To find that, let´s use the powerful UNITS CONTROL!

\(y\,\,{\text{miles}}\,\,\left( {\frac{{1\,\,{\text{h}}}}{{15\,\,{\text{miles}}}}} \right)\,\,\, = \,\,\frac{y}{{15}}\,\,{\text{h}}\)

Finally we go back to A´s real velocity/speed, using again UNITS CONTROL:

\(?\,\, = \,\,\,75\,\,\left( {\frac{{{\text{miles}}}}{{\text{h}}}} \right)\,\, \cdot \,\,\frac{y}{{15}}\,\,{\text{h}}\,\,\,\,{\text{ = }}\,\,\,{\text{5y}}\,\,\,{\text{miles}}\)


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

Regards,
fskilnik.
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Two cars travel on a highway in the same direction. If car A travels  [#permalink]

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New post 07 Sep 2018, 22:03
jennysussna wrote:
Two cars travel on a highway in the same direction. If car A travels at a rate of 75 mph and is \(y\) miles ahead of car B, which is traveling at 60 mph, in terms of \(y\), how many miles must car A travel to double the distance between itself and car B?


A) 3y
B) 4y
C) 5y
D) 6y
E) 8y

This problem is a "runaway." Faster car A starts at a "gap" distance of \(y\) miles ahead of B. Car A pulls away from B and until the \(y\)-mile gap between them doubles.

How far A must travel (to double \(y\)) is the key and the trap. \(y\) doubled is not the issue. 2y is not the answer.

The issue: Car A has to double \(y\) miles while B still moves during Leg 2.

One method: Choose a time traveled for Leg 1 to "assign" \(y\)

(1) Original gap of \(y\) miles?
• Assign a time based on relative speed
• SAME direction of travel? Subtract to get relative speed,
\(R=(Rate_{A}-Rate_{B})=(75-60)=15mph\)

• Let Leg 1 travel time = 2 hours
Car A traveled \((75*2) = 150\) mi
Car B traveled \((60*2) =120\) mi
Car A is \((150-120) = 30\) miles ahead of B
Starting gap distance \(y= 30\)

(2) Distance that A must travel to double \(y=30\)?
• start distance gap: 30 miles
• Desired distance gap: (30 * 2) = 60 miles
Car A travels (60-30) = 30 additional miles
• Car A does NOT cover 60 more miles. The 30-mile original gap never shrinks. It increases slowly, by 30 miles, to 60 miles

(3) Time for A to cover Leg 2 = 30 more miles? B still moves.
Use relative speed, R = 15 mph. D = 30
RT = D, so \(T=\frac{D}{R}\)
A's time: \(\frac{30mi}{15mph}=2\) hrs

(4) Distance that A traveled in Leg 2?
• (A's speed * A's time) = A's distance
• A's distance: \((75mph*2hrs)=150\) miles
• A traveled 150 miles to double the original gap

(5) \(150\) miles in terms of \(y=30\)? \(\frac{150}{30}=5\)
\(150=5y\)

Answer C
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Re: Two cars travel on a highway in the same direction. If car A travels  [#permalink]

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New post 10 Nov 2019, 15:23
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This is an expanding gap problem.
gap -> distance between two cars A & B
gap = y miles( given)

Question - Time taken by A to double the gap between A & B.
Find time taken by the gap , t= distance/ relative speed

t= y/(75-60) = y/15

time taken to cover y miles of gap is y/15.

Car A is already y miles apart, to double the gap , Car A needs to travel y/15 more hours.

distance travelled by Car A in y/15 hours = 75 * y/15 = 5y.

Ans: 5y
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Re: Two cars travel on a highway in the same direction. If car A travels  [#permalink]

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New post 10 Nov 2019, 22:28
jennysussna wrote:
Two cars travel on a highway in the same direction. If car A travels at a rate of 75 mph and is \(y\) miles ahead of car B, which is traveling at 60 mph, in terms of \(y\), how many miles must car A travel to double the distance between itself and car B?


A) 3y
B) 4y
C) 5y
D) 6y
E) 8y

(75-60)*t= Y
t=Y/15
Distance travelled by A in time t=75*Y/15
=5Y
C:)
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Re: Two cars travel on a highway in the same direction. If car A travels  [#permalink]

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New post 10 Nov 2019, 23:13
We know 3 facts.

A speed- 75 mph
B speed- 60 mph
Gap=Y

The trick is understanding the question is not asking us to find distance for distance traveled 2*Y but rather,
GAP+GAP=2*the gap. We already know the gap is "Y miles",we need to find distance A needs to travel to make another "Y miles" GAP.

So, the equation we can derive,

GAP=(Speed of A* Time)-(Speed of B*Time)

75T-60T=Y miles

T=Y/15 Miles
(Note: Time will be same for both the cars when they are "Y" miles apart)

Therefore,

Distance A Travels= (Time)*(Speed of A)

=75 mph * Y/15 = 5Y

This is a layman solution, you can solve this less than 2 mins as long as you get the trick.
Answer (C)
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Re: Two cars travel on a highway in the same direction. If car A travels   [#permalink] 10 Nov 2019, 23:13
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