QuestionQuestion

The following story poses an “arithmetical” problem. Your mission is to read it and answer the questions at the end.

EXCELSIOR.
“Goblin, lead them up and down.”

The ruddy glow of sunset was already fading into the sombre shadows of night, when two travelers might have been observed swiftly — at a pace of six miles an hour — descending the rugged side of a mountain; the younger bounding from crag to crag with the agility of a fawn, while his companion, whose aged limbs seemed ill at ease in the chain mail armor habitually worn by tourists in that district, toiled on painfully at his side.

As is always the case under such circumstances, the younger knight was the first to break the silence.

“A goodly pace, I trow!” he exclaimed. “We sped not thus in the ascent!”
“Goodly, indeed!” the other echoed with a groan. “We clomb it but at three miles in the hour.”
“And on the dead level our pace is—?” the younger suggested; for he was weak in statistics, and left all the details to his aged companion.
“Four miles in the hour,” the other wearily replied. “Not an ounce more,” he added, with that love of metaphor so common in old age, “and not a farthing less!”
“Twas three hours past high noon when we left our hostelry,” the younger man said, musingly. “We shall scarce be back by supper-time. Perchance mine host will roundly deny us all food!”
“He will chide our tardy return,” was the grave reply, “and such a rebuke will be meet.”
“A brave conceit!” cried the other, with a merry laugh. “And should we bid him bring us yet another course, I trow his answer will be tart!”
“We shall but get our deserts,” sighed the elder knight, who had never seen a joke in his life, and was somewhat displeased at his companion’s untimely levity. “Twill be nine of the clock,” he added in an undertone, “by the time we regain our hostelry. Full many a mile shall we have plodded this day!”
“How many? How many?” cried the eager youth, ever athirst for knowledge.

The old man was silent.
“Tell me,” he answered, after a moment’s thought, “what time it was when we stood together on yonder peak. Not exact to the minute!” he added hastily, reading a protest in the young man’s face. “An’ thy guess be within one poor half-hour of the mark, ’tis all I ask of thy mother’s son!
Then will I tell thee, true to the last inch, how far we shall have trudged betwixt three and nine of the clock!”
A groan was the young man’s only reply; while his convulsed features and the deep wrinkles that chased each other across his manly brow, revealed the abyss of arithmetical agony into which one chance question had plunged him.

1. State the “arithmetical” problem being posed in your own words.
2. Set up the “arithmetical” problem as an equation or equations to be solved.
3. Solve the arithmetical question. (With or without the equations from question.
Include a complete explanation of why your solution is correct.

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Q1 )There are two arithmetical problems. The first: how far did they walk. The second at what time T (hours past 3 pm) did they reach the top of the mountain.
Q2) We assume that they walk from the hostel (starting at 3 pm) on flat terrain, walk a distance F to the foot of the mountain at 4 mph, climb the mountain up a distance M at 3 mph.
Let T be the number of hours past 3 pm that they reach the top of the mountain:
T=F/4 +M/3.
Here the first term is the amount of time spent on flat terrain (time = distance / speed) and the second term is the time spent on climbing the mountain.

It is also given that the total time walking is 6 hours, so:
6=F/4+M/3+M/6 + F/4
Here the third term is the time spent descending the mountain at 6 mph. and the last term is the time spent on the return.
Curiously:
6=F/4+M/3+M/6 + F/4=F/2+M/2=(F+M)/2
The total distance traveled, D, is equal to:
D=F+M+M+F=2(M+F).

So the set of equations we have to work with is:
(1) T= F/4 +M/3.
(2) 6=(F+M)/2
(3) D=2(M+F).
Question: Determine D and determine T (if at all possible)...
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