Question 1198944: he lived 1/6 of his life as a child; 1/12 as a boy; 1/7 as an adolescent; 1/2 as a householder; and died as an old man for 9 years." Found 2 solutions by Alan3354, greenestamps:Answer by Alan3354(69443) (Show Source):
The wording is indeed awkward: "...died as an old man for 9 years".
The intended meaning is that the 9 years are what he had left after the other stages of his life.
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.
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
The total of those is 75 years; add the 9 years as an old man to get the correct total of 84.
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
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.
