How Old in 1900?

Date: 11/09/2002 at 17:34:14
From: Josh
Subject: Word Problem

In 1930, a correspondent proposed the following question: 
A man's age at death was one twenty-ninth of the year of 
his birth. How old was the man in 1900?

Date: 11/10/2002 at 12:47:46
From: Doctors Greenie and Ian
Subject: Re: Word Problem

Hi Josh, 

Since this problem was posed in 1930, the man must have died 
in 1930 or earlier.

The other date given in the problem is also significant.  
By asking what the man's age was in 1900, we know that he 
didn't die before 1900.

Suppose the man's age at death was x.  Then, according to 
the given information, the man's year of birth was 29x.   
He lived for x years after that, so the year of his death 
was (29x + x), or 30x.  

Only one year between 1900 and 1930 is divisible by 30, which 
is 1920.  So he must have died in 1920.

Again, we're told that "the man's age at death was 1/29th of 
the year of his birth".  If he died in 1920, and he lived x years,
then he must have been born in the year (1920 - x).  So

     (age at death) = (1/29)(year of birth)

                  x = (1/29)(1920 - x)

This will let you figure out his age when he died, which will 
let you figure out his age in 1900.  Can you take it from here?

- Doctors Greenie and Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/11/2002 at 05:29:13
From: Confused
Subject: Word Problem

Hi Dr. Math,

I was able to follow you up to getting the year of the man's 
death, but I can't see how you came up with his age at death, 
unless you just tried guessing a lot of different values.  
Is there a way to get it directly?

Date: 11/11/2002 at 16:41:36
From: Doctor Ian
Subject: Re: Word Problem

Hi,
 
Let's review what we know:

  1) The man lived for x years.
  2) He died in 1920.
  3) His age at death was 1/29th of the year of his birth.

From (1) and (2) we can conclude:

  4) He was born in (1920-x).

And now we're ready to make use of (3):

   (age at death) = (1/29)(year of birth)

                x = (1/29)(1920 - x)

              29x = 1920 - x

              30x = 1920

                x = 1920/30

                  = 64

So he died in 1920 at age 64.  That makes him 44 years old 
in 1900.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/