Problem: A Tangled Tale (click here to see the problem statement)

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My solution.

Faux Pas: I have to admit, I fell to the temptation to take the average of the given rates and multiply the result by 5 hours.
Once I got my head screwed on straight(er), here's what I did:

Recognizing that RATE is the RATIO between DISTANCE and TIME;

and that TIME can be determined by the ratio; and that the TOTAL TIME is ;

I constructed the spreadsheet below. Here I set up t-values for each leg of the trip, based on the given rates for each leg, and looked for sums that equaled 5 (hours). Note that the Distance for the fourth leg equals that of the first (Level Ground), and that the distances of the second and third legs (Uphill and Downhill) are equal. I started with the assumption that each leg had a distance of 1 mile, then adjusted the Uphill and Dowhnhill distances:

So with a Level Distance of 1 mile and an Uphill/Downhill Distance of 9 miles, given the rates of travel during each leg, resulted in a walk of 20 miles over 5 hours.

I then fixed the Level Distance at 2 miles and adjusted the Uphill/Downhill Distance. In this case, 2 miles on level ground and 8 miles Uphill and Downhill resulted in a 20 mile journey over 5 hours.

It seems that the total of the Level Ground mileage and the Uphill/Downhill mileage is always 10, and therefore, the total Distance is always 20 miles:

Examining the problem Algebraically.

As observed earlier, we are dealing with RATE, DISTANCE, and TIME,

where RATE is expressed by;

DISTANCE is expressed by;

and TIME is expressed by; .

I designated the overall RATE as

and the overall rate is also expressed by; .

But as the Level Distance remained constant on the first and fourth legs, then .

And the Uphill nd Downhill distances were equal, so .

Therefore, the overall rate is simplified to

and reduced to

or ,

as the total TIME was given to be 5 hours.

aaa

The Total TIME is

or

,

and given values for the TOTAL TIME and the RATE for each leg, we now have

.

But again, as and , then

.

So the Total Distance is 20 miles.

The overal RATE, though not asked for in the problem - I've gone this far, might as well keep going - is , 4 mph, that is.

a a a

PS: I leave for the reader to consider the meaning of the given situation if either the Level Distances or Uphill/Downhill Distance are ZERO.