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On a partly cloudy day, Derek decides to walk back from work 07 May 2018, 15:42
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Hi GMATPrepNow & VeritasPrepKarishma

I couldn't recognize that I can get S equal 2 from the beginning, so I used the weighted average as follows:

Let ds is the distance walked when it's sunny while dt is the total distance.

2.8 (Average speed) = S x ds/dt + (S+1)(dt-ds)/dt

Let ds/dt = K

2.8 = SK + (S+1)(1-X)
S = X + 1.8

The only answer choice that keeps S as an integer (2) is X = 1/5 => answer C

May I know what's wrong in my approach?
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On a partly cloudy day, Derek decides to walk back from work 07 May 2018, 16:49
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hisho
Hi GMATPrepNow & VeritasPrepKarishma

I couldn't recognize that I can get S equal 2 from the beginning, so I used the weighted average as follows:

Let ds is the distance walked when it's sunny while dt is the total distance.

2.8 (Average speed) = S x ds/dt + (S+1)(dt-ds)/dt

Let ds/dt = K

2.8 = SK + (S+1)(1-X)
S = X + 1.8

The only answer choice that keeps S as an integer (2) is X = 1/5 => answer C

May I know what's wrong in my approach?
Show more

Your solution assumes that average speed is based on the proportions of the distances traveled at each speed.
So, for example, if we traveled half the distance at 40 mph and traveled the other half the distance at 60 mph, then the average speed would be 50 mph.
However, this is not how average speeds work.

For more on this, watch the following video:

Cheers,
Brent
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KarishmaB
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On a partly cloudy day, Derek decides to walk back from work 07 May 2018, 20:38
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hisho
Hi GMATPrepNow & VeritasPrepKarishma

I couldn't recognize that I can get S equal 2 from the beginning, so I used the weighted average as follows:

Let ds is the distance walked when it's sunny while dt is the total distance.

2.8 (Average speed) = S x ds/dt + (S+1)(dt-ds)/dt

Let ds/dt = K

2.8 = SK + (S+1)(1-X)
S = X + 1.8

The only answer choice that keeps S as an integer (2) is X = 1/5 => answer C

May I know what's wrong in my approach?
Show more


Average Speed = Total Distance / Total Time

Average Speed = (Distance1 + Distance2) / (Time1 + Time2)

Average Speed = (Speed1*Time1 + Speed2*Time2) / (Time1 + Time2)

What you have done is

Average Speed = (Speed1*Distance1 + Speed2*Distance2) / (Distance1 + Distance2)
which is illogical.
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On a partly cloudy day, Derek decides to walk back from work 02 Jul 2018, 09:42
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On a partly cloudy day, Derek decides to walk back from work. When it is sunny, he walks at a speed of s miles/hr (s is an integer) and when it gets cloudy, he increases his speed to (s + 1) miles/hr. If his average speed for the entire distance is 2.8 miles/hr, what fraction of the total distance did he cover while the sun was shining on him?

A. 1/4
B. 4/5
C. 1/5
D. 1/6
E. 1/7
Show more

Let the total distance = d and x = the distance he covered when it was sunny. Thus, the distance he covered when it was cloudy is d - x. We need to determine the value of x/d.

Furthermore, we see that s must be 2. That is because if s = 1, then s + 1 = 2. We can see that if both speeds are 2 mph or less, then the average speed can never be 2.8 mph. A similar situation exists if s = 3 (or more), and s + 1 = 4 (or more). We can see that if both speeds are 3 mph or greater, then the average speed again can never be 2.8 mph. Therefore, s must be 2 since it given that it’s an integer. Therefore, his “sunny” speed is 2 mph, and his “cloudy” speed is 3 mph.

Since (total distance)/(total time) = average speed, we have:

d/[x/2 + (d - x)/3] = 2.8

d/2.8 = x/2 + (d - x)/3

Multiplying the equation by 84, we have:

30d = 42x + 28(d - x)

30d = 42x + 28d - 28x

2d = 14x

2/14 = x/d

1/7 = x/d

Answer: E
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On a partly cloudy day, Derek decides to walk back from work 25 Apr 2019, 17:42
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Don't fall for the trap! This problem is A LOT easier than many of the mathematical solutions in this forum seem to show. You just need to visualize what is happening, and use the leverage the problem gives you to strategically attack the question. It isn't about the math. Remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time (giving you more time for harder questions.) The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Let's talk strategy here. Here is the full "GMAT Jujitsu" for this question:

Let's start with the embedded leverage. The phrase "\(s\) is an integer" is what makes this entire problem tick. If the only two rates that Derek can travel are \(s\) and \(s+1\), then the average rate must be between \(s\) and \(s+1\). Since the question clearly states that his "average speed for the entire distance is 2.8 miles/hr," then \(s=2\) and \(s+1 = 3\).

Using a strategy I call in my classes "Weight Balancing", it is easy to determine the ratio between \(s\) and \(s+1\). The attached image below shows what it looks like. Since \(2.8\) is closer to \(3\), this means that the average is weighted towards the \(3\). The distances from the average give us the ratio.
Attachment:
WeightBalancing.png
WeightBalancing.png [ 11.6 KiB | Viewed 3455 times ]

Thus, Derek walks back at a ratio of \(0.8:0.2\) or \(4:1\). It is cloudy 4 times as much as it is sunny. (Sounds like Derek lives in on the Oregon Coast!) :)

Since we have the ratio of time, it is very easy to determine the distance, since Distance = Rate * Time.

The distance traveled while it is sunny is: \(D = RT = 2*1x = 2x\)
The distance traveled while it is cloudy is: \(D = RT = 3*4x = 12x\)
(Note: I am including the scaling factor, "\(x\)" in these calculations to show we are dealing with a ratio of unknown amounts, but as you will see, the scaling factor will disappear...)

Since the problem asks us for the fraction of the total distance that Derek covered while the sun was shining on him, this would be:
\(\frac{2x}{2x+12x}=\frac{2x}{14x} = \frac{1}{7}\)

The answer is "E".

Now, for those of you that are preparing to take the GMAT, let’s look back at this problem through the lens of strategy. Your job as you study isn't to memorize the solutions to specific questions; it is to internalize strategic patterns that allow you to solve large numbers of questions. This problem can teach us patterns seen throughout the GMAT. First, whenever the problem gives us leverage such as the word "integer", PAY ATTENTION. This is often a hint that you will be using logic as much as math. This solution also uses a strategy I call in my classes "Weight Balancing." The idea is simple: whenever you are given an "average" value between two groups, see if it might be useful to know the ratio between the groups. In the case of this problem, that ratio is what allows you to solve the problem quickly and efficiently without any unnecessary math. And that is how you think like the GMAT.
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On a partly cloudy day, Derek decides to walk back from work 10 Nov 2021, 05:10
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Hello,

Is the algebra good here?

I do not get how do we go from the 1st line to the second?

2.8 = (x+y) / (x/2 + y/3)
2.8x/2 + 2.8y/3 = (x+y)

In other terms, if we multiply : (x+y)/(x/2 + y/3) by (x/2 + y/3) do we obtain (x+y)?
Or is (x/2 + y/3)/ (x/2 + y/3)= 1 ?

Thanks

WholeLottaLove
On a partly cloudy day, Derek decides to walk back from work. When it is sunny, he walks at a speed of s miles/hr (s is an integer) and when it gets cloudy, he increases his speed to (s + 1) miles/hr. If his average speed for the entire distance is 2.8 miles/hr, what fraction of the total distance did he cover while the sun was shining on him?

Looking for the fraction of his total distance that was covered when the sun was shining on him. We're going to have to assign variables to sun distance and cloud distance so we can ultimately plug into the formula for average speed. We will also need to get the time taken for each leg of the journey which means we also need to know the speed traveled for each leg.
x=sun
y=clouds
The fraction of time spent walking when it is sunny = x/(x+y) (x divided by the total distance)

We have established that the speed for the first part = 2 and the speed for the second part =3

We are given average speed so it is fair to assume that we will apply it to a formula: Average speed = total distance/total time taken
Time taken:
sunny part of the journey: x/s (distance/speed)
cloudy part of the journey: x/s+1

2.8 = (x+y) / (x/2 + y/3)
2.8x/2 + 2.8y/3 = (x+y)
8.4x/6 + 5.6y/6 = x+y
8.4x+5.6y/6 = x+y
8.4x+5.6y = 6x+6y
2.4x = .4y
6x=y
x=1/6y
x/y=1/6

Answer: E. 1/7
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On a partly cloudy day, Derek decides to walk back from work Updated on: 08 Aug 2023, 04:30
Originally posted by KarishmaB on 30 Jan 2022, 22:26.
Last edited by KarishmaB on 08 Aug 2023, 04:30, edited 1 time in total.
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Naptiste
Hello,

Is the algebra good here?

I do not get how do we go from the 1st line to the second?

2.8 = (x+y) / (x/2 + y/3)
2.8x/2 + 2.8y/3 = (x+y)

In other terms, if we multiply : (x+y)/(x/2 + y/3) by (x/2 + y/3) do we obtain (x+y)?
Or is (x/2 + y/3)/ (x/2 + y/3)= 1 ?

Thanks

WholeLottaLove
On a partly cloudy day, Derek decides to walk back from work. When it is sunny, he walks at a speed of s miles/hr (s is an integer) and when it gets cloudy, he increases his speed to (s + 1) miles/hr. If his average speed for the entire distance is 2.8 miles/hr, what fraction of the total distance did he cover while the sun was shining on him?

Looking for the fraction of his total distance that was covered when the sun was shining on him. We're going to have to assign variables to sun distance and cloud distance so we can ultimately plug into the formula for average speed. We will also need to get the time taken for each leg of the journey which means we also need to know the speed traveled for each leg.
x=sun
y=clouds
The fraction of time spent walking when it is sunny = x/(x+y) (x divided by the total distance)

We have established that the speed for the first part = 2 and the speed for the second part =3

We are given average speed so it is fair to assume that we will apply it to a formula: Average speed = total distance/total time taken
Time taken:
sunny part of the journey: x/s (distance/speed)
cloudy part of the journey: x/s+1

2.8 = (x+y) / (x/2 + y/3)
2.8x/2 + 2.8y/3 = (x+y)
8.4x/6 + 5.6y/6 = x+y
8.4x+5.6y/6 = x+y
8.4x+5.6y = 6x+6y
2.4x = .4y
6x=y
x=1/6y
x/y=1/6

Answer: E. 1/7
Show more

Yes, Naptiste

\(\frac{a}{b} * b = \frac{ab}{b} = a\)

The b in the denominator get's cancelled with the b in the numerator.

So

\(\frac{(x+y)}{(x/2 + y/3)} * (x/2 + y/3) = (x + y)\)

Though the intent of this question is to use reason and ratios and avoid algebra.

With an average speed of 2.8 and two consecutive integer speeds, we know that the two speeds are 2 and 3.

If the average is 2.8, ratio of time taken at speed 2 and speed 3 = (3 - 2.8)/(2.8 - 2) = 1/4

(Check this post if you are not sure why: https://anaprep.com/arithmetic-weighted-averages/
and this video: https://www.youtube.com/watch?v=_GOAU7moZ2Q )

Say time taken was t and 4t.

Then ratio of distance covered at speed 2 and speed 3 = 2*t/3*4t = 1/6

So he covered 1/7th of the distance at speed 2 i.e. when the sun was shining on him.

Answer (E)
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On a partly cloudy day, Derek decides to walk back from work 17 Aug 2026, 07:14
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hi Bunuel,
if one had to back solve this, how would the approach be like?
also, some of the algebric approached has k in it, not sure what that denotes...
little help please

thanks,
Swetha
emmak
On a partly cloudy day, Derek decides to walk back from work. When it is sunny, he walks at a speed of s miles/hr (s is an integer) and when it gets cloudy, he increases his speed to (s + 1) miles/hr. If his average speed for the entire distance is 2.8 miles/hr, what fraction of the total distance did he cover while the sun was shining on him?

A. 1/4
B. 4/5
C. 1/5
D. 1/6
E. 1/7
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On a partly cloudy day, Derek decides to walk back from work 17 Aug 2026, 09:46
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SwethaReddyL
hi Bunuel,
if one had to back solve this, how would the approach be like?
also, some of the algebric approached has k in it, not sure what that denotes...
little help please

thanks,
Swetha

Show more

First note that the two speeds must be 2 mph and 3 mph.

Then backsolve the answer choices as the fraction of the distance traveled at 2 mph.

For example, try E, 1/7. Take the total distance as 7 miles:

Sunny: 1 mile at 2 mph, so time = 1/2 hour.
Cloudy: 6 miles at 3 mph, so time = 2 hours.

Total distance = 7 miles
Total time = 2.5 hours

Average speed = 7/2.5 = 2.8 mph.

So E works.

Answer: E.

For alternative solution check HERE.

Hope this helps.
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