How long was he walking?

Every day, Jack arrives at the train station from work at 5 pm. His wife leaves home in her car to meet him there at exactly 5 pm, and drives him home. 

One day, Jack gets to the station an hour early, and starts walking home, until his wife meets him on the road. They get home 30 minutes earlier than usual. How long was he walking? 

Distances are unspecified. Speeds are unspecified, but constant.

Give a number which represents the answer in minutes.

He must be walking for....minutes. Click to know! 

Jack's Walking Duration in Journey!

What was the question?

It's important to think from wife's point of view in the case.

Ideally, had Jack somehow informed earlier his wife about his 1 hour early arrival then his wife's (and his as well) total 60 minutes in round trip would have been saved. 

30 minutes of her round trip are saved which means only 15 minutes of each leg of her trip must have been saved. That is she meets her husband only 15 minutes earlier on the day instead of 60 minutes earlier (if Jack had informed her earlier). 

Hence, Jack must be walking for 45 minutes.

Let's understand this with example.

Suppose wife needs exactly one hour to reach the station every day. She leaves home at 4 PM everyday and reach station at 5 PM & drive Jack home at 6 PM

On one day, Jack arrived at 4 PM and started walking whereas wife leaves home at the same time as usual. They reach home at 5:30 PM.

30 minutes of wife saved indicates that she took 45 minutes to meet husband (instead of 1 hour) at 4:45 PM (instead of 5PM, only 15 minutes earlier) and took him to home at 5:30 PM (instead of 6PM) in next 45 minutes (instead of 1 hour) thereby saving 15 + 15 = 30 minutes only.

Since, Jack started walking at 4 PM and meet her wife at 4:45 PM, he must be walking for 45 minutes.   

Walking Up An Ascending Escalator

My wife and I walk up an ascending escalator. I climb 20 steps and reach the top in 60 seconds. My wife climbs 16 steps and reaches the top in 72 seconds. If the escalator broke tomorrow, how many steps would we have to climb?

Here is way to find the total number of steps! 

Total Steps on Broken Escalator

What was the question?

Climbing 16 steps makes you to reach at the top in 72 seconds but if you climb 20 steps then it requires 60 seconds only.

That means, climbing 4 steps extra pushes you at the top saving 12 seconds. This is the speed of the escalator which is 4 steps per 12 seconds.

That is escalator speed is 1 step per 3 seconds and 20 steps per 60 seconds. But in 60 seconds I climb 20 steps more hence total steps on escalator must be 40.

This is how escalator pushes me 20 steps + I climb 20 steps = 40 steps in 60 seconds.

On the other hand my wife climb 16 steps + escalator pushes her (40-16) i.e. 24 steps = 40 steps in 72 seconds.  


MATHEMATICAL APPROACH :

Let's suppose there are 'x' steps on escalator. If M is my speed , W is my wife's speed and E is speed of the escalator in terms of steps per second then,

M = 20/60 = 1/3 and W = 16/72 = 2/9.

In my case, for relative speed (steps per second) x/60, we have,

M - E = x/60

1/3 - E = x/60    ...........(1)

And in my wife's case, for relative speed (steps per second) x/72, we have, 
 
W - E = x/72

2/9 - E = x/72  ..............(2)

Subtracting (2) from (1),

(1/3) - (2/9) = (x/60) - (x/72)

(1/9) = x / 360

X = 40

So there are 40 steps on the escalator. 

Divide The Cake Into Equal Parts!

I have just baked a rectangular cake when my wife comes home and barbarically cuts out a piece for herself. The piece she cuts is rectangular, but it’s not in any convenient proportion to the rest of the cake, and its sides aren’t even parallel to the cake’s sides. 

 
I want to divide the remaining cake into two equal-sized halves with a single straight cut. How can I do it?



This is how it can be cut! 

Cutting The Cake Into Equal Parts!

What was the problem? 

Generally, a line drawn through the center of rectangle divides it into 2 equal parts.
Hence, a line drawn through the centers of both rectangles would divide each of them into 2 equal parts as shown below. (To get the center of each rectangle, all we need to do is draw diagonals of both).

A Mathematical Clue From The Merchant

A rich merchant had collected many gold coins. He did not want anybody to know about them. 

One day, his wife asked, “How many gold coins do we have?”

After pausing a moment, he replied, “Well! If I divide the coins into two unequal numbers, then 32 times the difference between the two numbers equals the difference between the squares of the two numbers.”

The wife looked puzzled. Can you help the merchant’s wife by finding out how many gold coins they have?

Here are mathematical steps to find those!

Using The Mathematical Clue

What was that clue?

Since when divided into 2 unequal numbers difference won't be 0. Let x and y be the 2 unequal numbers.

As per merchant,

32 (x - y) = x^2 - y^2

32 (x - y) = (x - y) (x + y)

Dividing both sides by (x - y) which is non zero as x is not equal to y,

32 = x + y

x + y = 32.

Let's verify with x = 30 and y = 2. So 32 (x - y) = 32 ( 30 - 2) = 896. And x^2 - y^2 = 30^2 - 2^2 = 900 - 4 = 896.

Hence, Merchant had 32 coins in total.