When was Jack born?

Question

Definition: There are 10 decades in a century. When n=1, the n-th decade implies the first decade (years 0-9) of a century or a life.

Jack was in his x-th decade of life in the y-th decade of the 20th century. In the x-th decade of the 21st century he would be in his y-th decade of life. x+y happens to be the sum of the digits of his age on his birthday in 2021. In what year was he born?

Answer 1

New try,

1900+10y-10x = decade of birth

2000+10x-10y = decade of birth

On equating both the equations we get:

y = 5+x

So possible candidates are in (y,x) = {(5,0),(6,1),(7,2)}

Now converting it to decade system. the solution becomes (6,1) (7,2) and (8,3)

(6,1) cannot be solution because then his age would be in between 71-79 in 2021 and no number between this range equals 7

(7,2) so his age would be between 71-79 in 2021, but only 72 is the age which equals 9, so it is our solution.

(8,3) so his age would be between 71-79 in 2021, but only age 74 is the age which equals 11, so we also include it in our solution.

So my guess is that it is 72 and 74.

so his year of birth can be either

2021-72 = 1949 or

2021-74=  1947.

Comments

No, not 1948, which is in the 5th decade of the 20th century. Can you show your working?

Decades are numbered as ordinals not cardinals.

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Sir, I posted my incorrect solution. Please have a look at it.

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I think there are two possible answers, but neither is 1948 or 1939. If Jack was born in 1948 he would be 73 in 2021; and if he was born in 1939 he would be 82 in 2021. In each case the sum of the digits is 10, so we know x+y=10, which gives (x,y)=(1,9), (2,8), (3,7), (4,6), (5,5), (6,4), (7,3), (8,2), or (9,1).

All of these can be eliminated because he would be too young or too old. If, for example, he’s in his 20s (3rd decade) in the 1960s (7th decade), then he has to be in his 60s in the 2020s. That means he has to be between 60 and 69. But in 2021 he’ll be 73 or 82 if he was born in 1948 or 1939.

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Sir,

Did you come up with the problem on your own, or did you find it somewhere else?

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I invented it one day when I was walking the dog.

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Your equation in lines 5-6 is incorrect because it doesn’t take into account his age in the two centuries. 1900+10y represents a decade in the 20th century but doesn’t relate that decade to his age. Neither does 2000+10x. Think it through: start with an expression that works out the decade in which he was born, then you can work out an equation using that birth decade to relate x and y. This equation will lead you fairly quickly to the right answer. Any one of the two possible answers would be acceptable as a solution. You don’t have to find both, although that would be a bonus!

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Sir, is this correct?

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Yes, you got the right equations at last! You also got two years that I missed. I corrected a small error at the end of your solution, because I think it was a simple mistake.

I found another couple of solutions, which are different from yours, but, hey, what does it matter? The most important thing was knowing how to get the answer. Well done!

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Can you show your solution?

And I request you to give some calculus problems even if it is not invented by you. Please.