How did they get the numerator and denominator in this beast of a question?

Question

The question starts with this poem and I've been able to work it to the point at the bottom. My question is in the last equation at the bottom of this post, how on Earth did they come up with 25 for the numerator and 28 for the denominator? I have tried everything I can think of to figure it out and I'm stuck. Please help!

"Here lies Diophantus," the wonder behold . . .
Through art algebraic, the stone tells how old:
"God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage
After attaining half the measure of his fathers life
chill fate took him.
After consoling his fate by this science of numbers
for four years, he ended his life."

Find Diophantus' age at death.

My first task is to "translate" the poetry from the headstone into practical terms:
"Boyhood" can stand for pre-adolscent childhood; he spent one-sixth of his life in this period. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved
"Youth while whiskers grew" can stand for pubescence (the teenage years, into young adulthood); he spent one-twelfth of his life in this period.
"Ere marriage began" can stand for "unmarried adulthood" or "bachelorhood"; he spent one-seventh of his life in this period.
He had five years between the wedding and the time his first child was born.
Tragically, this child died young, living only half as long as his father eventually would; looked at the other way, half of Diophantus' life was spent while his child was alive.
Diophantus died four years after burying his child.




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I will let d stand for Diophantus' age at death. Then:

childhood: d/6
adolescence: d/12
bachelorhood: d/7
childless marriage: 5
age of child at death: d/2
life after child's death: 4

His whole life had been divided into intervals which, when added together, give the sum of his life. So I'll add the lengths of those periods, set their sum equal to his (as-yet unknown) total age, and solve:

d/6 + d/12 + d/7 + 5 + d/2 + 4 = d
( 25/28 )d + 9 = d

Answer 1

I don't like these kinds of problems, and I'll tell you why: The author clearly was thinking outside of the box but did not take into account the objective nature of mathematics versus the subjective nature of poetry and creative writing. For example, where you see that "d/4" should be the age of the child, I read the poem as:

 

"After attaining half the measure of his fathers life
chill fate took him.

After consoling his fate by this science of numbers
for four years, he ended his life."

 

To mean: "[(d-4)/2] + 4"

 

But it can be read either way. Of course this changes the problem drastically depending on how one interprets the poem.

 

Here is another potential discrepancy:

 

"God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife."

 

To which subject from the first line does "one-twelfth more" refer? Is it one-twelfth more of the full lifespan, or one-twelfth more of the boyhood, which was one-sixth of the full lifespan?

 

If the poem is read in this alternative way, we actually have a series of terms, with each consecutive term being a multiple of the preceding term (i.e. boyhood is d/6, youth is (d/6)*(1/12), etc.).

Answer 2

This “beast of a question” obviously took some skill (both math and poetic) to construct.

You should realize, that translating word (story) problem into concise, precise math notation is pretty easy when you make a table of word translations.  For example,
     Of                    usually means                            multiply
     One-sixth                                                              1/6
     More, then, in five, after, in, after                  add
     Half of  d                                                              (1/2)*d

This makes “translating” very easy:
     d/6  +  d/12  +  d/7  +  5  +  d/2  +  4  =  d
     (  d/6 + d/12 + d/7 + d/2) + 9  = 12            [grouping the terms that have d]

 

Now, the math part of the problem.  Find the common denominator so we can add fractions.

Finding prime factors is an easy way to do this:
      6 has prime factors     2,3
     12  has prime factors    2,2,3
      7   has prime factors       7
      2 has prime factors       2

The HCF has prime factors   2,2,3,7  (the smallest set that contains all of the above)
       And    2 * 2 * 3 * 7   = 84

So,
          d * (14 +7 + 12 + 42) / 84   +   9   = d
          (25/28)d  +  9  =  d
Then
         9  =  3d/28    now, divide both sides by 3
        3  = d/28       then, cross-multiply
         84 = d           answer

To check:
    Is  84/6  + 84/12  + 84/7  + 5  + 84/2  + 4  =  84    ?      (you check it out)

p.s., no problem is safely complete until you have checked your work