On a road trip, Kip drove one-quarter the distance at an average speed

Question

On a road trip, Kip drove one-quarter the distance at an average speed of v miles per hour, one-quarter the distance at an average speed of 2v miles per hour, one-quarter the distance at 4v miles per hour, and one-quarter the distance at 8v miles per hour. In terms of v, what was Kip’s average speed (in miles per hour) for the entire trip?

A) 15/4v
B) 15v/4
C) 32v/15
D) 4v/15
E) 15v/32

*kudos for all correct solutions

A) 15/4v
B) 15v/4
C) 32v/15
D) 4v/15
E) 15v/32

*kudos for all correct  solutions

pushpitkc · Accepted Answer

Let the total distance traveled be 4d miles. So each quarter of the total distance is d miles.

We are given the average speeds of the 4 quarters - v,2v,4v, and 8v miles/hr

The time taken to cover the individual distances are  \frac{d}{v},\frac{d}{2v},\frac{d}{4v},\frac{d}{8v}

The total time taken is  d(\frac{1}{v} + \frac{1}{2v} + \frac{1}{4v} + \frac{1}{8v}) = \frac{d(8+4+2+1)}{8v} = \frac{15d}{8v}

Therefore, Kip’s average speed (in miles per hour) for the entire trip is  \frac{4d}{\frac{15d}{8v}}  =  \frac{32v}{15}   (Option C)

Abhishek009 · Answer

Let the Total distance be 12miles

Kip drove 3 miles at v miles/hr , so time taken is 3/v hr
Kip drove 3 miles at v miles/hr , so time taken is 3/2v hr
Kip drove 3 miles at v miles/hr , so time taken is 3/4v hr
Kip drove 3 miles at v miles/hr , so time taken is 3/8v hr

So, Total time taken is 3/v + 3/2v + 3/4v + 3/8v = 24+12+6+3/8v = 45v/8

Thus, average speed will be 12*8/45 = 32v/15 , Answer must be (C)

SSM700plus · Answer

We all know  speed = \frac{distance}{time}

Let distance be x. Kip drove one-quarter each time and exactly 4 times. distance traveled each time is  \frac{x}{4} .

Total Time taken = Time taken for Quarter 1 + Time taken for Quarter 2 + Time taken for Quarter 3 + Time taken for Quarter 4
 =  \frac{x}{4v} + \frac{x}{4*2v} + \frac{x}{4*4v} + \frac{x}{4*8v} 
 =  \frac{15x}{4*8v}

We also know that Avg Speed =  \frac{Total Distance}{Total Time} 
 =  \frac{x}{15x/4*8v} 
 =  \frac{32v}{15}

Ans - C

generis · Answer

Assign a value for speed  v , then choose a convenient number for distance, D.

v  increases by powers of 2. I want integers. 
Let   v=1 
 2v=2 
 4v=4 
 8v=8

D=8  for each leg will yield an integer for the time for all four legs.

Time,  t=\frac{D}{v}

Time,  Leg_1 :  \frac{8}{1}=8  
Time,  Leg_2 :  \frac{8}{2}=4 
Time,  Leg_3 :  \frac{8}{4}=2 
Time,  Leg_4 :  \frac{8}{8}=1

Total   T: (8+4+2+1) = 15  
Total   D: (8*4) = 32

Ave speed =  \frac{TotalD}{TotalT}=\frac{32}{15}

Use  v=1 . We need the answer choice that yields  \frac{32}{15}

Immediately eliminate   A, B, and E  . Wrong denominator. 
(Per Answer A, 4 is not a factor of 15. Nor does (4*1) = 15)

C) 32v/15 =     \frac{32}{15}   KEEP

D) 4v/15 =     \frac{4}{15}   REJECT

Answer C

BrentGMATPrepNow · Answer

Since we aren't told the total distance, let's  assign a nice value  to the distance. 
Let's say  32  miles = TOTAL distance. 
So, each quarter =  8  miles

Average speed = total distance/total travel time 
So, we must find the travel time for EACH QUARTER and add them together.

Time = distance/speed, so: 
Time spent driving v mph =  8 /v = 8/v
Time spent driving 2v mph =  8 /2v = 4/v
Time spent driving 4v mph =  8 /4v = 2/v
Time spent driving 8v mph =  8 /8v = 1/v

TOTAL travel time = 8/v + 4/v + 2/v + 1/v 
=   15/v  
 
 Average speed = total distance/total travel time 
=  32 /(  15/v  )
= ( 32 )(  v/15  )
= 32v/15

Answer: C

Cheers,
Brent

VipulGarg80 · Answer

Alternative short approach :

Avg speed of 1st two and next two given speed shall be  (2ab/a+b) . ( since, distances covered are equal.)

= 2*v*2v/3v and 2*4v*8v/12v 
= 4v/3 and 16v/3
again, finding the avg speed of two obtained avg speeds:
= (2 * 4v/3*16v/3) / 20v/3
 Hence,  32v/15 .

bumpbot · Answer

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On a road trip, Kip drove one-quarter the distance at an average speed

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