A Long Journey of 27000 Miles

The MacDonalds are planning a long car journey of 27,000 miles. If they use tires that last 12,000 miles each, how many tires will they need, and how can they make the best use of them?

This is how usage of tires can be optimized!

Optimizing The Use of Tires in Long Journey!

How long the actual journey is?

Since each tire would be traveling 27,000 miles, when car travels 27000 miles; tire -miles are equal to 27000 x 4 = 108000.

But the each tire lasts for 12000 then tires required = 108000/ 12000 = 9.

Now managing use of these 9 tires need some planning.

For first 12000 miles of 27000 miles they can use 4 out of 9 tires. For the rest of 15000 miles use of remaining 5 tires need to be planned.

Best way is to change 1 tire after every 3000 miles like below.

First 3000 miles: Tires 1, 2, 3, 4

Second 3000 miles: Tires 2, 3, 4, 5

Third 3000 miles: Tires 3, 4, 5, 1

Fourth 3000 miles: Tires 4, 5, 1, 2

Fifth 3000 miles: Tires 5, 1, 2, 3

This way, it's made sure that each tire is used for only 12000 miles. And this is how the journey can be completed using 9 tires only.

The Race Between 2 Brothers

Zachary challenges his brother Alexander to a 100-meter race. Alexander crosses the finish line when Zachary has covered only 97 meters.

The two agree to a second race, and this time Alexander starts 3 meters behind the starting line.

 
If both brothers run at the same speed as in the first race, who will win?

He will win the second race for sure! 

Source 

Winner of The Race Between 2 Brothers

What was the race of?

Let's assume that, Alexander completes the first race in time 't'. That means, he reaches at the finish line after running 100m after time 't' since start of the race. In the same time interval, Zachary could reach only 97m.

Now, in second race too, Alexander covers 100m once again in time interval 't' & Zachary runs 97m distance. Since, Alexander started 3m ahead of start line, at this point of time both are at the same point with 3m left to complete the race. 

 
Since, Alexander had won first race with faster speed & speed of both are unchanged in second race, it's clear that Alexander will take less time to cover leftover 3m distance. Hence, Alexander will be winner of the second race. 

MATHEMATICAL APPROACH:

Let's suppose Alexander took 10s to complete the first race. Then, his speed is 100/10 = 10m/s.

In 10s, the Zachary could run only 97m. So, his speed is 97/10 = 9.7m/s.

In the second race, their respective speeds are unchanged but Alexander has to run 103m to reach at the finish line compared to 100m of Zachary.

Hence, time taken by Alexander to reach at the finish line = 103/10 = 10.3s and that taken by Zachary = 100/9.7 = 10.30929s.

It's clear that Zachary needs more time to finish this race too. Hence, Alexander will be the winner in this race as well. He beats Zachary by 100 - (10.3x9.7) = 0.09m.

Truth Tellers and Liars in Circle

On a certain island there live only knights, who always tell the truth, and knaves, who always lie.

One day you find 10 island natives standing in a circle. Each one states: "Both people next to me are knaves!"

Of the 10 in the circle, what is the minimum possible number of knights?

Do you think there can be 5?

Identifying Number of Truth Tellers in Circle

What was the task given?

Recalling the given situation. 

On a certain island there live only knights, who always tell the truth, and knaves, who always lie.

One day you find 10 island natives standing in a circle. Each one states: "Both people next to me are knaves!"

 
Every Knight must be surrounded by 2 Knaves and every Knave has to be surrounded by at least one knight to satisfy the given condition. So there must be Knave-Knight-Knave groups must be standing in circle. 

Now if Knave of previous group is counted for the next group, then there will be 5 knights in the circle as shown below.

But the question asks minimum possible number of Knights in the circle.

So after forming 3 groups of Knave-Knight-Knave separately (total 9 in circle), the last person will be obviously surrounded by 2 knaves. Hence, he must be Knight. See below.

This way there can be only 4 knights standing in the circle without violating the given condition.
 

Cars Around Interesting Circular Track

Around a circular race track are n race cars, each at a different location. At a signal, each car chooses a direction and begins to drive at a constant speed that will take it around the course in 1 hour. When two cars meet, both reverse direction without loss of speed. Show that at some future moment all the cars will be at their original positions.

Here is that proof!


Source