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Age-Old Age Problem

Teresa is 13-years-old, and twelve children (including Teresa) were at her birthday party--four children from each of three families, the Jones, the Smiths, and the Bensons. Each child was of a different age and together represented twelve of the numbers 1 to 13. Being a mathematics teacher, Teresa's mother added the ages of the children in each family and noted these interesting results:

  • JONES Family: Total of ages is 41 years and includes the 12-year-old
  • SMITHS Family: Total of ages is 22 years and includes the 5-year-old
  • BENSONS Family: Total of ages is 21 years and includes the 4-year-old
Oops, almost forgot to tell you that only the Jones family had children who were born in consecutive years.

What are the ages of the children in each family?


Hint: This problem reverberates with the idea of setting up systematic cases, and testing various options. Do you feel the vibrations?


Solution Commentary: I will get you started...subtract off the three known ages for each family, so now we have the remaining ages of 1,2,3,6,7,8,9,10,11,13 and the Jones total of 29, while Smiths and Bensons both total 17. Now, with the remaining numbers, can you get a sum of 29 without using either the 11 or 13? How does your response to this question connect to the "oops" part of the problem's statement?

So, you now have some cases to test for the Jones family: _ _ 11 12 or _ _ 12 13, with new "magic sums" of 18 or 16 respectively. Chose the first case...By playing with the positioning of the 13 (which is required as it is Terersa's age), can that age fit in either the Smiths or Bensons families?

The remainder of the problem is left to you, as it involves checking possibilities against all of the clues or requirements (the "oops" plays a big role in this process).

At the end, do you get a unique answer, or is more than one answer possible?

Some people like this type of problem...while some do not. I am not sure what this means.