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What was average speed challenge?
A man drives 1 mile to the top of a hill at 15 mph. That means he took, 1/15 hours i.e.4 minutes to reach at the top of a hill.
To achieve average speed of 30 mph, the man has to complete 2 miles trip in 1/15 hours i.e. 4 minutes. But he has already taken 4 minutes to reach at the top of a hill, hence he can't achieve average speed of 30 mph over entire trip.
MATHEMATICAL PROOF:
Let 'x' be the speed needed in the journey down the hill.
Average Speed = Total Distance/Total time
Average Speed = (1 + 1)/(1/15 + 1/x)
30 = 2/(1/15 + 1/x)
(1/15 + 1/x) = 2 / 30 = 1/15
1/x = 0
x = Infinity/Not defined.
To conclude, it's impossible to achieve average speed of 30mph in trip.
A father wants to take his two sons to visit their grandmother, who
lives 33 kilometers away. His motorcycle will cover 25 kilometers per
hour if he rides alone, but the speed drops to 20 kph if he carries one
passenger, and he cannot carry two. Each brother walks at 5 kph.
Can the
three of them reach grandmother’s house in 3 hours?
Do you think it's impossible? Click here!
What was the challenge in the journey?
Yes, all three can reach at grandmother's home within 3 hours. Here is how.
Let M be the speed of motorcycle when father is alone, D be the speed of motorcycle when father is with son and S is speed of sons. Let A and B are name of the sons.
As per data, M = 25 kph, D = 20 kph, S = 5 kph.
1. Father leaves with his first son A while asking second son B to walk. Father and A drives for 24 km in 24/20 = 6/5 hours. Meanwhile, son B walks (6/5) x 5 = 6 km.
2. Now father leaves down son A for walking and drives back to son B. The distance between them is 24 -6 = 18 km.
3. To get back to son B, father needs 18/(M+S) = 18/(25+5) = 18/30 = 3/5 hours & in that time son B walks for another (3/5) x 5 = 3 km. Now, son B is 6 + 3 = 9 km away from source where he meets his father. While son A walks another (3/5) x 5 = 3 km towards grandmother's home.
4. Right now father and B are 24 km while A is 6 km away from grandmother's home. So in another 24/20 = 6/5 hours father and B drive to grandmother's home. And son B walks further (6/5) x 5 = 6 km reaching grandmother's home at the same time as father & brother B.
In this way, all three reach at grandmother's home in (6/5) + (3/5) + (6/5) = 3 hours.
In this journey, both sons walks for 9 km spending 9/5 hours and drives 24 km with father taking (6/5) hours. Whereas, father drives forward for 48 km (24 km + 24 km) in (6/5) + (6/5) hours and 15 km backward in 3/5 hours.
You find yourself trapped at top an 800 foot tall building. The surrounding land is completely flat, plus there
are no other structures nearby. You need to get to the bottom,
uninjured, and can only safely fall about 5feet.
You look down the four walls; they are all completely smooth and featureless, except that one of the walls has a small ledge 400feet above the ground. Furthermore, there are two hooks, one on this
ledge, and one directly above it on the edge of the roof. The only tools
you have are 600feet of rope, and a knife.
How do you get to the bottom?
This should be your strategy!
Why strategy needed to be planned?
1.Tie one end of the rope to the to hook and climb down to the
ledge.
2. Cut (without dropping) the rope that hangs below the ledge, then
climb back to the roof carrying the extra rope that you cut. You now have two lengths of rope: one that is 400 feet long and one
that is 200 feet long.
3.At the top,
untie the rope from the hook.
Now setup the ropes like : Tie a small loop at one end of the 200-foot long rope.
String the 400-foot long rope through the loop so that half of its
length is on either side of the loop. Make sure that the loop is snug
enough that the 400-foot long rope won't fall out by itself, but loose
enough that you can pull the rope out later.
4. Now, tie the end of the 200-foot rope without the loop to the first
hook. The 200-foot long rope lets you climb halfway to the ledge.
5.For
the remaining 200 feet, you carefully climb down the 400-foot rope,
which hangs down 200 feet from where it is held by the loop.
6.Once you
get to the ledge, pull the 400-foot rope out of the loop.
7. Finally, tie it to the
second hook, and climb the rest of the way to the ground.
A story tells that, as a 10-year-old schoolboy, Carl Friedrich Gauss was asked to find the sum of the first 100 integers. The tyrannical schoolmaster, who had intended this task to occupy the boy for some time, was astonished when Gauss presented the correct answer, 5050, almost immediately.
How did Gauss find it?
Actually, he used this trick!
