# DIOPHANTO'S AGE God allowed him to be boy for the sixth part of his life, and, with the addition of a twelfth part, the cheeks clothed with hair. After...

DIOPHANTO'S AGE

God allowed him to be boy for the sixth part of his life, and, with the addition of a twelfth part, the cheeks clothed with hair. After another seventh part of his life to God he lit the torches wedding, and five years after marriage granted him a son. Alas! poor son was born too late, after reaching half of the entire life of his father, the chill Fate took him. Having consulate with the science of numbers for four more years, he ended his life.

How was Diophantus at Death?

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steveschoen | Certified Educator

Hi, oldnick.  I hope this assist.

Taking one part at a time, we don't know how old Diophontus is, so we mark that as x:

x = Diophantus' age

So, 1/6 of that is "x*1/6" or "x/6".  Then, we add 1/12 his age, or "x/12".  Then, we add 1/7 of his age, or x/7.  So far, we have:

x/6 + x/12 + x/7

Then, we add 5 years.  Then, Dio has a son that lives half of Dio's life.  Dio lives x years, so his son lived x/2 years.  So, we add x/2.  Then, we add 4 more years to complete Dio's life.  So, we have:

x/6 + x/12 + x/7 + 5 + x/2 + 4

Now, this is suppose to represent Dio's entire life.  So, this has to be equal to x.  So:

x/6 + x/12 + x/7 + 5 + x/2 + 4 = x

Various ways to solve from here.  Simplifying a little bit, we can combine the x/6, x/12, and x/2, exactly the same as 1/6, 1/12, and 1/2, which is 9/12, or 3/4.  So, "x/6 + x/12 + x/2 = (3/4)x".  So, the equation becomes:

(3/4)x + x/7 + 9 = x

Subtracting (3/4)x from each side:

x/7 + 9 = x/4

Multiplying each side by the common denominator, 28, we have:

4x + 252 = 7x

Subtracting each side by 4x:

252 = 3x

Dividing by 3:

x = 84

So, Diophantus was 84 years old at Death.  Checking, 1/6 of his life would be 14 years.  1/12 of his life would be 7 more years, 21 years old.  Adding 1/7 of his life would be 12 more years, or 33 years old, when he got married.  5 more years when he had his son, 38 years old.  His son lived half as long as he did entirely.  Half of 84 is 42 years.  So, 38+42= 80 years old.  4 more years to his death, 80+4 = 84 years old.

I hope this helps, oldnick.  Good luck.