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

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Intern
Joined: 24 May 2018
Posts: 6
Two cars travel on a highway in the same direction. If car A travels  [#permalink]

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Updated on: 07 Sep 2018, 20:23
8
00:00

Difficulty:

65% (hard)

Question Stats:

55% (01:53) correct 45% (02:15) wrong based on 40 sessions

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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, 12:32.
Last edited by generis on 07 Sep 2018, 20:23, edited 2 times in total.
Edited and formatted the question, renamed the topic
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 480
Re: Two cars travel on a highway in the same direction. If car A travels  [#permalink]

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07 Sep 2018, 14:08
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1
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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07 Sep 2018, 21:03
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 $$y=30$$? $$\frac{150}{30}=5$$
$$150=5y$$

Two cars travel on a highway in the same direction. If car A travels &nbs [#permalink] 07 Sep 2018, 21:03
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