Problematics | Wage hikes

In an age when social media influencers are seemingly earning loads of money, you will occasionally come across individuals who invested in education and are now wondering if they made the right choice. If that startled you, let me assure readers: Problematics is not venturing into career advice or any scholarly comparison between the relative merits of studying and influencing. The only reason for bringing this up is that such misgivings have a context in this week’s puzzle.

The idea that other people earn disproportionately more, in fact, predates the influencer age. People have always wondered why skills they perceive as inferior to their own fetch higher salaries. British mathematician Ian Stewart, a puzzler and author who popularises mathematics, provides clinching proof that people who know more must earn less. The proof, which I have slightly rewritten, goes like this:

Given: Knowledge is Power, and Time is Money.

But, Power = Work/Time

=> Time = Work/Power

=> Money = Work/Knowledge

Therefore: for a fixed amount of work, the more you know, the less money you get.

(Source: Professor Stewart’s Hoard of Mathematical Treasures)

Ask any worker in any factory. They will always tell you that the supervisor knows less than they do but still earns more for the same amount of work. An illustrative example of worker dissatisfaction appears in my puzzle below, which I have crafted after drawing inspiration from a Henry Dudeney classic that had nothing to do with workers.

#Puzzle 140.1

Years ago, three workers at a factory were grumbling about their pay. “Do you know the supervisor gets twice as much as the three of us put together?” said Worker #1. “Well, you get more than Worker #2 or I,” said Worker #3. Worker #1 consoled his colleagues: “I have been here longer than you, and newer workers must naturally get less. You know everyone starts as an apprentice without pay for a year before they are confirmed at a monthly wage of ₹1000 a month, which becomes ₹2000/month the next year and keeps rising by ₹1000 per month every successive year. Unfortunately, that means even the supervisor will keep getting that hike.”

Five years later, the three workers were discussing the same topic along with a fourth colleague (who hadn’t yet joined the factory at the time of the previous conversation). “Do you know the supervisor gets as much as the four of us put together?” said Worker #4. The other three nodded, glum as ever.

Ten years have passed since the second conversation, and it is now today. The four workers have stuck on with the company, grumbling a little less than before because of the yearly hikes of ₹1000/month. Today, they are holding yet another adda, and this time it includes a fifth worker, who is newer than the others and therefore grumbles about everyone.

“Hey Worker #1, you get as much as Worker #4 and I together,” says Worker #5.

“Your time will come,” Worker #1 says.

Worker #4, being neither old nor new, is sitting on the fence. “At the time Worker #5 joined, you (addressing Worker #1) were earning the sum of Worker #3’s salary and mine,” Worker #4 says.

“Both of you earn more now,” says Worker #1 who, with ambitions high, is waiting for the supervisor to retire.

Worker #2 tries to change the subject: “The supervisor earns half as much as all five of us put together.”

“Look who’s talking. You (addressing Worker #2) and Worker #1 together earn as much as the other three of us put together,” says Worker #3.

Worker #2 shoots back: “Do you want me to put that another way, Worker #3? Context matters.”

If you can work out the experience of each worker and the supervisor in years (as of today), their present salaries will automatically follow. Also, try and explain what Worker #2 meant by that last cryptic remark.

#Puzzle 140.2

Two sisters are celebrating their fifth birthday. They are sisters because they were born to the same parents. It is their fifth birthday because both of them are 5 years old day, having been born on the same day of the same month in the same year. But they are not twins.

How is that possible?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 139.1A

Hi,

The escalator walked by the professor has 48 steps. Let the total number of steps be x, the speed of the escalator be e . So x= 16 + 16e. Also x= 24 + 12e (because the total number steps actually covered by the professor is 12 x 2=24 when she is taking 2 steps at a time). Solving the two equations gives e=2, so that x= 16 + 32= 48 steps.

— Ajay Ashok, Delhi

#Puzzle 139.1B

Hi Kabir,

The escalator walked by the Lecturer has 100 steps. Let the escalator take y steps in time t during which the lecturer takes 50 steps. So the total number of steps is 50 + y. When he takes 125 steps at 5 times the original speed, the time taken is (125/50) x (1/5)t = t/2. During this time the escalator moves y/2 steps, so the total number of steps is 125 – y/2. Equating 50 + y = 125 – y/2 and solving, we get y = 50. Therefore, the total number of steps is 50 + y = 100.

— Sabornee Jana, Mumbai

#Puzzle 139.1C

In my original version of the third escalator puzzle, I had both walkers moving in the same direction. Then I had misgivings (just like some people we discussed earlier as being overawed by what social media influencers earn) and revised the puzzle to have the two walkers moving in opposite directions. As it turns out, both versions work; some readers have solved one version and some the other.

Same direction: The escalator has 300 steps. If the number of steps in the escalator are x, the escalator moves (x – 150) steps in the time it takes Ms Ahmed to reach the top. Similarly, the escalator moves (x – 75) steps in the time Mr Basu takes to reach the top. Since Ms Ahmed takes 3 steps in the time Mr Basu takes one, the ratio of the time taken = (75/1)/(150/3) = 3/2. This gives the equation 3/2(x – 150)= x - 75; or x= 450- 150 = 300.

— Kanwarjit Singh, Chief Commissioner of Income tax, retired

Opposite directions: The escalator has 120 steps. Suppose the number of steps is E. If it moves m steps with each stride of Mr Basu, then m/3 is the number of steps the escalator moves with each stride of Ms. Ahmed. Therefore, for Ms Ahmed, E = 150 – 150 (m/3); and for Mr Basu, E = 75 + 75m. Equating and solving gives m = 3/5 and E = 120.

— Professor Anshul Kumar, Delhi

#Puzzle 139.2

Dear Kabir,

Given that the elder one of Professor Pandit’s two children is a daughter, there are two possibilities: GG and GB. So the probability that both are daughters is 1/2. Given that at least one of Lecturer Lalaji's two children is a boy, the possibilities are GB, BG and BB. So the probability that both children are boys is 1/3.

— Y K Munjal, Delhi

Given that were five sub-puzzles in two sets last week, it is not possible to prepare a separate list for one. So let’s limit this to two lists, those who have solved all five and those who have solved some of them.

Solved all five puzzles: Ajay Ashok (Delhi), Professor Anshul Kumar (Delhi), Aishwarya Rajarathinam (Coimbatore), Dr Sunita Gupta (Delhi), Shishir Gupta (Indore), Vinod Mahajan (Delhi), Yadvendta Somra (Sonipat)

Solved some of them: Sabornee Jana (Mumbai), Kanwarjit Singh, (Chief Commissioner of Income tax, retired), YK Munjal (Delhi), Sampath Kumar V (Coimbatore)

Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com.