Question 1198944
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The wording is indeed awkward: "...died as an old man for 9 years".<br
The intended meaning is that the 9 years are what he had left after the other stages of his life.<br>
For a quick setup using logical reasoning, note that all the numbers of years in the problem have to be whole numbers.  Then, since he lives 1/12 of his life as a boy and 1/7 of his life as an adolescent, his age when he died has to be a multiple of both 7 and 12.  The least common multiple of 7 and 12 is 84; and any larger common multiple of 7 and 12 would not be a reasonable age.<br>
So his age at death was 84.  Then he spent...
1/6 of 84 = 14 years as a child;
1/12 of 84 = 7 years as a boy;
1/7 of 84 = 12 years as an adolescent; and
1/2 of 84 = 42 years as a householder<br>
The total of those is 75 years; add the 9 years as an old man to get the correct total of 84.<br>
Of course you can set up the problem for solving using formal algebra:
(1/6)x + (1/12)x + (1/7)x + (1/2)x + 9 = x<br>
To solve that formally, you would multiply both sides of the equation by the least common multiple of the denominators.  But after that the required work will do exactly what I did above in the informal solution.<br>