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Math puzzle




Samantha had three children : Ellery, Jacob and Jesse.
Their combined ages totaled half of her age.
Five years later, during which time Evelyn was born, Samatha's age equalled the total of  all her children's ages.
Ten more years have now passed.
Delphine was born during that time.
At that event Ellery was as old as Jessy and Evelyn together.
The combined ages of all the children are now double Samantha's age, which is only equal to that of Ellery and Jacob together.
Ellery's age also equals that of the two daughters.

Question : How old are they?


  How to arrive at the answer? During the 5 yrs, if Evelyn is born, it could be any number < 5, isn't it? I am clueless as to how to solve this puzzle.


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Six people are mentioned. They are (with today's age in parentheses):
Samantha (Sa), Ellery (EL), Jacob (Ja), Jesse (Je), Evelyn (Ev) and Delphine (De)
Five times are mentioned. They are, in chronological order:
15 years ago; x years ago (when Ev=x was born); 10 years ago; y years ago (when De=y was born); and now.

So there's a timeline, with ages, that looks like this:


Sa-15        Sa-10              Sa     
El-15        El-10   El-y       El
Ja-15        Ja-10              Ja
Je-15        Je-10   Je-y       Je
      Ev-x=0 Ev-10   Ev-y       Ev=x     10<Ev<15
                     De-y=0     De=y      0<De<10

Using Ev=x and De=y, these six equations come from the OP:

  1. Sa = 2(El+Ja+Je)-75   
  2. Sa = (El+Ja+Je+Ev)-30 
  3. El = (Je+Ev)-De          
  4. Sa = (El+Ja+Je+Ev+De)/2
  5. Sa = El+Ja
  6. El = Ev+De

Because the ages are integers, we can deduce that

Sa, El, and De are ODD
Ja, Je, and Ev are EVEN

2. and 5. give Je+Ev = 30. Because 10<Ev<15 then 15<Je<20.
3. and 6. give Je = 2De. So 15<2De<20, or 7.5<De<10. So De=8, 9 or 10.
Since De is ODD, De=9. It follows that Je=18 and Ev=12.
From 3. El=21.
From 1. and 5. Ja=24
From 5. Sa=45.

But 1. gives Sa=51. So I made an error, which I will find, perhaps, tomorrow. (It's 5:00am now.)

But anyone reading this, feel free to check my work and post the right numbers. B))

This was a fun puzzle.

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