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

## 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.

##