On a partly cloudy day, Derek decides to walk

BTGModeratorVI wrote: ↑

Fri Jun 05, 2020 11:40 am

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

Answer: E
Source: Veritas Prep

Let's assign a nice value to the total distance traveled.

If Derek's average speed is 2.8 mph, then let's say that he traveled a total of 28 miles.
At an average rate of 2.8 mph, a 28 mile trip will take 10 hours.

Since Derek's average speed is BETWEEN 2 mph and 3 mph, we can conclude that Derek walked 2 mph when it was sunny, and he walked 3 mph when it was cloudy.

Let's t = number of hours walking while sunny
So, 10 - t = number of hours walking while cloudy

We'll begin with a word equation: (distance traveled while sunny) + (distance traveled while cloudy) = 28
Since distance = (speed)(time), we can now write:
(2)(t) + (3)(10 - t) = 28
Expand: 2t + 30 - 3t = 28
Solve: t = 2
In other words, Derek walked for 2 hours while sunny.

At a walking speed of 2 mph, Derek walked for 4 miles while sunny.
So, Derek walked 4/28 of the total distance while the sun was shining on him.
4/28
= 1/7

Answer: E

Cheers,
Brent

BTGModeratorVI wrote: ↑

Fri Jun 05, 2020 11:40 am

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

Answer: E

Solution:

Since the average speed for the entire distance is 2.8 mph and s and s + 1 are consecutive integers, the value of s must be 2. That is, his walking speed is 2 mph when it’s sunny and 3 mph when it’s cloudy. (Notice that s can’t be any integer other than 2 because 2.8 is between 2 and 3. For example, if s = 3, his two walking speeds will be 3 mph and 4 mph, respectively. However, that will never average to 2.8 mph for the entire distance.)

Now we can let 2 miles be the distance when he walks 2 mph and x miles be the distance when he walks 3 mph and create the equation:

Total distance / total time = average speed

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

2 + x = 2.8(1 + x/3)

2 + x = 2.8 + 2.8x/3

6 + 3x = 8.4 + 2.8x

0.2x = 2.4

x = 12

We see that the total distance is 2 + 12 = 14 miles and the fraction of the total distance he covered while the sun was shining on him is therefore 2/14 = 1/7.

Answer: E