During a trip, Francine traveled x percent of the total distance at an

I like to begin with a word equation:
Average speed = (total distance)/(total time)
For this question, let's let the total distance = D

Next, observe that: total time = (time spent driving 40 mph) + (time spent driving 60 mph)

time spent driving 40 mph = distance/speed
Aside: distance driven = (x/100)(D)
So, time spent driving 40 mph = (x/100)(D)/40

time spent driving 60 mph = distance/speed
Aside: if x% of the distance was driven at 40 mph, then the distance driven at 60 mph = [(100-x)/100](D)
So, time spent driving 60 mph = [(100-x)/100](D)/60

Here comes the awful algebra ...

Total time = (x/100)(D)/40 + [(100-x)/100](D)/60

Simplify ...
Total time = xD/4000 + [100D-xD]/6000
Total time = 3xD/12000 + [200D-2xD]/12000
Total time = (xD+200D)/12000

And finally,
Average speed = (total distance)/(total time)
= D/[(xD+200D)/12000]
= (12000D)/(xD+200D)
= (12000)/(x+200)

Answer:

RELATED VIDEO

Answer: option E

Check solution attached

Hi All,

We're told that During a trip, Francine traveled X percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. We're asked, in terms of X, what was Francine's AVERAGE SPEED was for the entire trip. This question can be approached in an number of different ways - and can be solved rather easily by TESTing VALUES.

IF....
Total Distance = 100 miles and X = 40....

40% of 100 = 40 miles, so Francine traveled 40 miles at 40 miles/hour --> 1 hour traveled
The rest = 60 miles, so Francine traveled 60 miles at 60 miles/hour --> 1 hour traveled

Total Distance traveled = 100 miles
Total Time traveled = 2 hours
Average Speed = 100/2 = 50 miles/hour

So, we're looking for an answer that equals 50 when X = 40. There's only one answer that matches...

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Can someone help with one aspect:

In other posts, people have been saying that the average speed is 48 m.p.h by showing this math:

(2)(40)(60)/(40+60)

How is 2 X 40 X 60 the total distance? Where do the individual numbers come from?

Thanks

Hi mborski,

In this question, the Average Speed will depend on the total distance that the car travels at each of the two speeds (40 miles/hour and 60 miles/hour). The ONLY time that the Average Speed will be exactly 48 miles/hour is when the car travels the SAME distance at each of those two speeds. Under all other situations, the Average Speed will be something OTHER than 48 miles/hour.

GMAT assassins aren't born, they're made,
Rich

Hi Bunuel,

2 Questions:
(1) Can we use the weighted average approach to solve this question? If so, how would we do that?
(2) Why can't we use the p1A1 + p2A2 = A(total) approach for this question?

Thank you!

Hi pruekv

Please remember: The average Speed is NOT the average of two speeds and it's a function of total distance and total time

For detail please watch the video here.
Video Time - 2:26 onward

why do we assume x=50?

Hi Elnur1997,

This is a reference to TESTing VALUES, which is a really useful Tactic in the Quant section of the GMAT. Since "X" refers to the percent of the distance traveled at 40 miles/hour (with the remaining distance traveled at 60 miles/hour), using X = 50 is an easy way to 'split' the distance into 2 pieces.

Based on the way that the overall question is written, there's a really easy way to answer the question with this same strategic approach. Make sure to note what the question is asking for: we're asked, in terms of X, what was Francine's AVERAGE SPEED was for the entire trip.

IF....
Total Distance = 100 miles and X = 40....

40% of 100 = 40 miles, so Francine traveled 40 miles at 40 miles/hour --> 1 hour traveled
The rest = 60 miles, so Francine traveled 60 miles at 60 miles/hour --> 1 hour traveled

Total Distance traveled = 100 miles
Total Time traveled = 2 hours
Average Speed = 100/2 = 50 miles/hour

So, we're looking for an answer that equals 50 when X = 40. There's only one answer that matches...

GMAT assassins aren't born, they're made,
Rich

Put x = 0, you should get 60 because the entire distance was travelled at 60 mph. Only option (C) and (E) are possible.
Put x = 50, half the distance would be travelled at 40 and half at 60 so avg speed will be 2ab/(a+b) = 2*40*60/(40 + 60) = 48
Out of (C) and (E) only option (E) gives 48.

Answer (E)

Elnur1997

You use x = 50 because you are assuming that 50% of the distance (half) was travelled at 40 mph and the other 50% at 60 mph.
We know how to calculate avg speed when two equal distances are travelled at speeds of a and b. The avg speed is given by the formula 2ab/(a+b)

Check this post for more: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2015/0 ... -the-gmat/

Video solution from Quant Reasoning starts at 15:35
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1

Generally on the GMAT:
When the same distance is traveled at two different speeds, the average speed for the entire trip will be a little closer to the lower speed than to the higher speed.
The reason:
It takes longer to travel the distance at the lower speed than at the higher speed.
Since more time is spent at the lower speed, the average speed for the whole trip will be closer to the lower speed.



Let x=50%, implying that half the distance is traveled at 40 mph and the other half at 60 mph, with the result that the average speed for the whole trip will be a bit closer to 40 than to 60.
Implication:
When x=50 is plugged into the correct answer, the result will be a bit less than 50.

A. \(\frac{(180-x)}{2} = \frac{180-50}{2} = 65\)
B. \(\frac{(x+60)}{4} = \frac{50+60}{4} = \frac{110}{4} = 27.5\)
C. \(\frac{(300-x)}{5} = \frac{300-50}{5} = \frac{250}{5} = 50\)
D. \(\frac{600}{(115-x)} = \frac{600}{115-50} = \frac{600}{65}\) = less than 10
Eliminate A, B, C and D.



E. \(\frac{12,000}{(x+200)} = \frac{12,000}{50+200} = \frac{12,000}{250} = \frac{1200}{25} = 48\)
Only E yields a value a bit less than 50.

Hi BrentGMATPrepNow, does total distance can only be 100? As if I use 120 but getting a different answer below and not sure why is that? Thanks Brent

120/ x/40 + 120-x/60
120/ (3x+240-2x)/120
120 * 120/ x+240
14400 / x+240

Be careful. In my solution, D = the total distance.
The 100 does not represent a distance.
The 100 is the "cent" (100 in French) part of x perCENT, as in "Francine traveled x percent of the total distance at an average speed of 40 miles per hour"

Thanks BrentGMATPrepNow, to clarify so D can only be 100 here due to x perCENT of D ? Thanks Brent

I don't believe I ever stated that D = 100. Can you highlight that part of my solution?

In the meantime...

Key property: k% = k/100
This means x percent = x/100

Given: Francine traveled x percent of the total distance at an average speed of 40 miles per hour
So, if D = the TOTAL distance driven, we can say that (x/100)D = the distance driven at 40 mph