A driver completed the first 20 miles of a 40-mile trip at an average

SOLUTION

A driver completed the first 20 miles of a 40-mile trip at an average speed of 50 miles per hour. At what average speed must the driver complete the remaining 20 miles to achieve an average speed of 60 miles per hour for the entire 40-mile trip? (Assume that the driver did not make any stops during the 40-mile trip.)

(A) 65 mph
(B) 68 mph
(C) 70 mph
(D) 75 mph
(E) 80 mph

\(average \ speed=\frac{total \ distance}{total \ time}=\frac{40}{total \ time}=60\). This implies that for the average time to be 60 miles per hour, the total time must be 40/60 = 2/3 hours.

Now, the first 20 miles were covered in (time) = (distance)/(speed) = 20/50 = 2/5 hours.

Thus, the remaining 20 miles should be covered in 2/3 - 2/5 = 4/15 hours, which means that the remaining 20 miles should be covered at an average speed (distance)/(time) = 20/(4/15) = 75 miles per hour.

Answer: D.
_________________

Average speed = (total distance)/(total time)
We already know that the total distance travelled = 40 miles
And we know that we want the average speed to be 60 miles per hour
So, our equation becomes: 60 = 40/(total time)
We can rearrange this equation to get: total time = 40/60 = 2/3 hours

During the first part of the trip, the driver travels 20 miles at a speed of 50 mph
Time to complete first part = distance/rate = 20/50 = 2/5 hours

During the second part of the trip, the driver travels 20 miles at an unknown speed. So let's say that speed is x mph
Time to complete second part = distance/rate = 20/x = 20/x hours

At this point we have enough information to create the following equation: 2/5 + 20/x = 2/3
To eliminate the fractions we'll multiply both sides of the equation by 15x to get: 6x + 300 = 10x
Subtract 6x from both sides to get: to get: 300 = 4x
Solve: x = 300/4 = 75

Answer: D

RELATED VIDEO

_________________

Video solution from Quant Reasoning starts at 13:44 here:

_________________

Answer = (D) 75 mph

Time = Distance / Speed

Setting up equation as shown in fig

LET X=20 MILES

x/50+x/y=2x/60
=> 1/y=1/30-1/50=1/75
=>y=75

HENCE D.

P.S.: You are doing direct average/ weighted average of speed, thats wrong. you need to check that the time it takes to cover the two individual 20 miles trip should be equal to the total time its takes to cover 40 miles with average speed 60 mph.

Hope this helps...!!

avg speed = total distance/total time

t1 = 20/50h = 0.4h
t2 = 20/x h

60 = 40/(0.4 + 20/x)

x= 75

Since the distance is same in both stretches of the journey,therefore average speed is Harmonic mean of the speed

Average speed = 2uv/(u + v)

Average speed = 60 ,
U or V = 50

60 = 2*50*v/(50 + v)
3 = 5v/(50 + v)
150 + 3v = 5v
2v=150
v = 75

Hence D

Basically, the first 20 miles took 24 minutes, so the second 20 miles need to take 16 minutes in order for the average to be 60miles/h..


We need to pick an option from A-E (which we call X), that in the denominator makes 120/(16*x) = 1. Note that the value needs to be divided by 10 before it is multiplied by 16.. The only value that works is D (16 * 7.5 = 120), and thus D is the answer.


Don't even ask me how I came to solve it with this convoluted mumbo jumbo but my brain worked on full gear and at the time that I did this it made perfect sense, even though Im not that good at explaining the whole process in hindsight.

You can use the "normal" distance/rate approach.

First, divide the trip:
For the whole trip he has to take 40 minutes (2/3 h) because he is driving at an average speed of 60m/h.
So first 20 miles at 50 m/h means that he takes 24 min (2/5 h) for half the distance.
This means that he has to take 16 minutes = 16/60 = 4 / 15 h for the last 20 miles. This gives us the equation:

distance = rate(x) * time
20 = x * 4/15
20*15/4 = x
300/4 = x
75 = x

Answer D.

why are you setting your problem to 40 miles in 40 mins? Where did 120 come from?

If you cant explain ANYTHING why write an explanation?

It's an instinctive method you often use when you learn to play with numbers in your head.
You want the average speed to be 60 miles/hr i.e. you need to cover 60 miles in 60 mins which means you must cover 40 miles in 40 mins.
The first 20 miles were covered at an average speed of 50 mph i.e. time taken (in mins) to cover the first 20 miles is 20/50 * 60 = 24 mins
You need to cover 40 miles in total 40 mins and you have already taken 24 mins during the first 20 miles. This means, you need to speed up now and cover the rest of the 20 miles in the leftover 16 mins.
What will be your speed in mph if you cover 20 miles in 16/60 hrs?
Speed = 20/(16/60) = 75 mph
_________________

A driver completed the first 20 miles of a 40 miles trip at an average speed of 50 miles per hour. At what average speed must the driver complete the remaining 20 miles to achieve an average speed of 60 miles per hour for the entire 40-miles trip? ( Assume that the driver did not make any stops during the 40-miles trip)

A. 65
B. 68
C. 70
D. 75
E. 80

Let \(x\) be the average speed during remaining 20 miles

Total Trip Time = Time to cover first 20 miles + time to cover remaining 20 miles
Since total trip distance is twice that of first part of the trip, we may write above equation as

\(\frac{1}{50} + \frac{1}{x}= \frac{2}{60}\)

Or,\(\frac{1}{x} =\frac{2}{60} - \frac{1}{50}\)
Or, \(x = \frac{300}{4}=75\) miles/hr

Answer: (D)

Speed = Distance/ Time

Time elapsed for 20Miles = (20/50) *60 = 24 Min
Remaining distance = 20 Miles

Remaining time:
@ 60 Miles/ hr, 40 Miles would take - 40 Minutes

So remaining time = 40-24 = 16 Mins

Speed required to cover 20 miles in 16 mins = (20/16)*60 = 75 Miles/hr

Average speed concept always confuses me. Anyways there is a formula

\(Avg. Speed = \frac{2(S_1)(S_2)}{S_1+S_2}\)


\(S_1\) being one speed and \(S_2\) being another

By substituting the value you will get 75 miles/hour


Answer D
Time Taken - 1:45
Difficulty level - 600

Can we use this formula:

60 = 20/50 + 20/x ?? If yes, what is the way to solve for x ?

NO, you can't use this Principle. In fact the only principle for calculating Average Speed is

Average Speed = Total Distance / Total Time

Solving Equation for you:

Here, Average Speed = 60
Total Distance = 40
Total Time = Time taken in Travelling 1st 20 Miles + Time taken in Travelling 2nd 20 Miles = (20/50) + (20/x) [Because Time = Distance/Speed]

i.e. 60 = 40/ [20/50 + 20/x]

i.e. 60 = 40/ 20[1/50 + 1/x] = 2/ [(x+50)/(50x)] = 2*50x / [(x+50)] = 100x / [(x+50)]

i.e. 60*[(x+50)] = 100x

i.e. 60x + 30000 = 100x

i.e. 40x = 3000

i.e. x = 300/4 = 75 mph

I hope It clears your Doubt!
_________________

We can use the following formula:

average rate = (distance 1 + distance 2)/(time 1 + time 2)

where average rate = 60, distance 1 = distance 2 = 20, time 1 = distance 1/rate 1 = 20/50 = 2/5, and time 2 = distance 2/rate 2 = 20/r (where r is the average speed of the remaining 20 miles).

Let’s now determine r:

60 = (20 + 20)/(2/5 + 20/r)

60 = 40/(2r/5r + 100/5r)

60 = 40/[(2r + 100)/5r]

60 = 200r/(2r + 100)

60(2r + 100) = 200r

120r + 6000 = 200r

6000 = 80r

r = 6000/80 = 600/8 = 75

Answer: D
_________________

The total distance is 40 miles, and we want the average speed to be 60 miles per hour.
Average speed = (total distance)/(total time)
So, we get: 60 = (40 miles)/(total time)
Solve equation to get: total time = 2/3 hours
So, the TIME for the ENTIRE 40-mile trip needs to be 2/3 hours.

The driver completed the first 20 miles of a 40-mile trip at an average speed of 50 miles per hour.
How much time was spent on this FIRST PART of the trip?
time = distance/speed
So, time = 20/50 = 2/5 hours

The ENTIRE trip needs to be 2/3 hours, and the FIRST PART of the trip took 2/5 hours

2/3 hours - 2/5 hours = 10/15 hours - 6/15 hours
= 4/15 hours
So, the SECOND PART of the trip needs to take 4/15 hours


The SECOND PART of the trip is 20 miles, and the time is 4/15 hours
Speed = distance/time
So, speed = 20/(4/15)
= (20)(15/4)
= 75 mph

Answer: D

Cheers,
Brent
_________________

VeritasPrepKarishma

I figured out. Sorry!

It's like:

t1/t2=(x-60)/(60-50)

Where t1= 2/5 & t2= 20/x

The equation looks something like this after solving a bit
2x/5*20=(x-60)/10
x=5(x-60)
X=5x-300
4x=300
x= 75

Option D