Jack and Jill went up a... mountain?

Probability Level pending

Jack and Jill wanted to fetch 2 pails of water. But they found out that the nearby wells were dry, and the only available well was on a tall mountain. Because they only had one pail, they decided that Jack starts climbing ahead of Jill, while she gets another pail. Jack, at a speed of 3kph, starts go up. Half an hour later, Jill comes back and starts to go up at a rate of 2kph. Once Jack had gotten to the top, he found out that the well there is defective, and it takes 30 minutes to fill each pail. He starts filling it up, and once it was full, he starts going down at the same rate. He meets Jill in the middle of the climb 2 1 2 2\frac{1}{2} hours after he started going up. He also finds that Jill is still going up once they meet.

Let the distance of one mountain climb be x x kilometers, and the distance Jack needs to walk going down after meeting Jill be y y kilometers.

What is x + y x + y ?

Note:

"One mountain climb" means going from the top to bottom or vice versa.

Meaning to say, Jack and Jill each made two mountain climbs (if we were to wait for the two of them to finish): one for going UP, and one for going down.

Also, assume that they can constantly walk at this rate, non-stop.


The answer is 9.

Renz Sibal by Renz Sibal , 17, Philippines

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1 solution

Renz Sibal
Dec 12, 2017

First, let's find the distance of one mountain climb, or x . x.

Since we are talking about distances and speeds, it would be nice to remember the formula:

d i s t a n c e = s p e e d × t i m e distance = speed \times time

It is given that Jack and Jill meet at one point after Jack started going up, where Jack is already going down and Jill is still going up.

If this is the case, then we can take their meeting point and the time Jack started going up as a reference.

First, for Jack, he has been walking since the time he started climbing minus the time he took to get water. That is:

time since he started walking - time he took getting water = total time spent walking

2 1 2 \frac{1}{2} hours - 1 2 \frac{1}{2} hours = 2 hours

So if he has been walking for 2 hours, he walked a distance of:

2 hours × \times 3 kph = 6 km

Jack has walked 6 km before meeting Jill.

As for Jill, she has been walking continuously half an hour after Jack started to do so.

2 1 2 \frac{1}{2} hours - 1 2 \frac{1}{2} hours = 2 hours

So, if she had been walking for 2 hours, she walked a distance of:

2 hours × \times 2 kph = 4 kph.

Jill walked 4 km before meeting Jack.

The two of the walked a distance of 6 + 4 = 10 6 + 4 = 10 km.

Take note that before they met, they made a total of 2 mountain climbs. Jack made one whole mountain climb plus the part he made before meeting Jill, and Jill did the remaining before meeting Jack.

So if we are finding the distance for 1 mountain climb, that is 10 2 = 5 k m = x \frac{10}{2} = 5 km = x

Now, we need y y , which is the remaining distance before Jack gets down. He walked a total of 6 km, and a mountain climb is 5 km. By now, He is going down, and he has cleared 6 - 5 = 1 km. He needs to go 5 - 1 = 4 km more. So, 4 = y 4 = y

We are being asked for x + y . x + y. x = 5 , y = 4. x + y = 123 x = 5, y = 4. x + y = \boxed{123}

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