# An easy mathematical question?

Asked by flutherother (28340) August 26th, 2010

Two travelers spend from 3 o’clock till 9 in walking along a level road, up a hill, and back home again: their pace on the level being 4 miles an hour, up hill 3, and down hill 6.

Find the distance walked: also (within half an hour) the time of reaching the top of the hill.

Sounds easy, but can you find the answer?

Observing members: 0 Composing members: 0

This does not “sound easy.”

Every time a math question is presented to me I’m pretty sure my brain shuts down by default.

ANef_is_Enuf (25194)

How the hell can we answer this if we don’t know what percentage of the road is level, and what percentage is hill?

Seek (34729)

All right, maybe it doesn’t sound easy but it does have an answer.

flutherother (28340)

The hill could be ten feet high. That’s a totally different calculation than a two-mile high hill.

Messing with dyscalculic people… making me think it’s my fault this question doesn’t have an answer… dammit.

Seek (34729)

@flutherother it might sound easy. I am just exceptionally bad at math.

ANef_is_Enuf (25194)

46

CMaz (26228)

Sorry 46 doesn’t work.

flutherother (28340)

Okay, since you won’t give it to me, I’ll make it up myself. The uphill portion is 1.5 miles long.

So, we have six hours of walking.

Minus the ½ hour of walking at 3 mph, and the 15 minutes of walking at 6 mph.

5¼ hours of walking at 4 mph is 21 miles.

21/2 is 10.5.

So the answer is 11.5 miles from the beginning of the road to the top of the hill.

The lack of necessary information in the OP is astounding.

Seek (34729)

Good try! but then he would have walked 10.5 miles on the flat and 1.5 miles of hill which is 12 miles.

flutherother (28340)

24 miles, and top of hill after 3.5 hours….????

harple (10441)

@flutherother I calculated that. Try again.

Your question plainly states that you wanted to know how far it was to the top of the hill.

Seek (34729)

Harple, you have the first bit right as did kolinahr (apart from the arithmetic). The total distance walked is 24 miles. The next question is when did he reach the top of the hill (to within half an hour)

flutherother (28340)

Okay, the total time walked is 6 hours. The pattern of walking is level, uphill, downhill, level.
That translates to 4mph, 3mph, 6mph, 4mph. If tth hill is 3 miles, up and down will take him 1–½ hours, leaving 4–½ hours to cover the flat in both directions. If he walks for 4–½ hours at 4 mph, he will be covering 18 miles total, or 9 miles flat, 3 miles uphill, 3 miles downhill, 9 miles flat.

2.25 + 1 = 3.25 hours to the top of the hill, or 6:15

BarnacleBill (16065)

I beg your pardon! Top of hill at 6.30 pm?

harple (10441)

6:30 is not correct I’m afraid but 24 miles total walking is correct

flutherother (28340)

I thought you said this was easy.

Frenchfry (7564)

Hmmm… not even if the flat is 6 miles, and the hill 6 miles?...

harple (10441)

See, the thing is, no matter what, we have to come up with part of the equation out of thin air.

Maybe if you gave the whole damned question someone could get the answer?

Seek (34729)

The key is in the relationship between distance and time. The hill is the same distance up and down, but the speed up is 2 times as long as the time down. The speed for the flat sections is ⅔ that of the downhill speed, and you walk the distance twice. You walk for 6 hours and the total distance is 24 miles. Regardless, to the top of the hill is 12 miles.

I need a pencil and a piece of paper…

BarnacleBill (16065)

Did @flutherother tell you already that one of the travelers is carrying the other on his back?

rebbel (26755)

I’m going to take a wild stab in the dark & say it was Jack & Jill.They came straight down because Jack was a clumsy bugger…..am I right? What do I win?

ucme (45780)

Okay if the distance walked is 24 miles, then the actual distance covered is half that because there are two walkers. The top of the hill is actually at 6 miles distance from the start rather than 12 miles.

BarnacleBill (16065)

I really don’t have the faintest idea how to even begin working this out.

downtide (23475)

OK I’ll put you out of your misery. Harple was right so sorry about that. The thing is that the time it takes to walk a mile up and then a mile down the hill is exactly the same time it takes to walk 2 miles on the level that is half an hour. In six hours the traveller will cover 24 miles whether it is all on the flat or all up and down the hill.

You donâ€™t know if the journey is all flat with a tiny hill at the end or all mountain with a tiny distance of flat ground in front of it so you can look at these two extremes.

If it is all flat the traveller will turn around after walking for three hours that is at 6:00pm
If it is all mountain the traveller will have to turn around after walking for 4 hours that is at 7:00pm

If you say 6:30 as Harple did then you have got the answer to within half an hour and you are correct

flutherother (28340)

Even when the answer is presented to me on a plate, I still can’t understand how it’s calculated.

downtide (23475)

By the way this puzzle is quite old and was created by Charles Dodgson, or Lewis Carroll the author of Alice in Wonderland. He was a mathematician and made up lots of these puzzles.

flutherother (28340)

How can the hill be level and up hill?

mowens (8353)

x = miles traveled up hill = miles traveled down hill

y = miles traveled level

Total distance traveled = 2x + y

Now what we know is:

Time for walking uphill = x/3
Time for walking downhill = x/6
Time for walking in along the level road = y/4

What we know is that the total time is 6 which means: x/3 + x/6 + y/4 = 6, so x/2 + y/4 = 6

Multiply both sides by 4 and you get: 2x+y = 24

BonusQuestion (1482)

Let L be the time required to walk to the hill and D the time required to walk down the hill. Since they ascend at half the speed at which they descend, the time spent in ascending the hill is double that spent descending or 2*D and the total time on the hill is 3*D. The distance up the hill and back is 3*2*D + 6*D = 12*D. Since 12*D = 4*(3*D), they would have gone up and down the hill in the same amount of time had they traveled at a rate of 4 miles per hour, so the trip was 4*6 or 24 miles. If they started at the base of the hill (L=0), they would have traveled 12 miles at a rate of 3 miles per hour, thus taking 4 hours to arrive at the top at 7:00. On the other hand, if the hill was flat (D=0), they would have traveled 12 miles at a rate of 4 miles per hour, thus taking 3 hours to reach the top at 6:00. Thus, they arrived at the top at 6:30, give or take half an hour.

ratboy (15157)

WHAT??!? how is that easy?!? i swear my brain cant handle maths