How old are the children?
Problem
A student in a maths class was trying to get some information from her teacher.
| Sally : | "How many chidren have you got, Mrs Smith?" |
| Mrs Smith: | "Three" |
| Sally: | "How old are they?" |
| Mrs Smith: | "If you multiply their ages together, it comes to 72" |
| Sally: | "That doesn't give us enough information, Mrs Smith" |
| Mrs Smith: | "If you add the ages you get the number of this room" |
| Sally: | "That still doesn't give us enough information, Mrs Smith" |
| Mrs Smith: | "The youngest is called Amanda" |
| Sally: | "Thank you. Now we know how old they are !" |
Well, how old are they?
Student Solutions
There was a good postbag for this question.
All submissions got as far as listing the possible sets of three numbers with product 72. There are 12 such "triples", namely
[1, 1, 72], [1, 2, 36], [ 1, 3, 24], [1, 4, 18],
[1, 6, 12], [1, 8, 9], [2, 2, 18], [2, 3, 12],
[2, 4, 9], [2, 6, 6], [3, 3, 8] and [3, 4, 6].
But what about the door number? How does Mrs Smith's answer help Sally?
This part of the solution was a stumbling block for many entrants.
We assume that Sally knew the number. Now all the triples have a different sum EXCEPT [2,6,6] and [3,3,8] which both sum to 14. So the door number must be 14, otherwise the information would have told Sally the answer. It follows that the correct triple must be one of these two.
Finally, since Amanda is the youngest child, she is not a twin, and the correct triple is [2, 6, 6] rather than [3, 3, 8], so we know that Mrs Smith has a 2 year old called Amanda and twins aged 6.
There were many partially correct solutions and plenty of correct answers with explanations that were incomplete or plain false, but solutions from the following were both complete and correctly argued.
Jessica, Daniel and Mark of Jack Hunt School, Peterborough;
Alice, Ashley, Alice and Elisabeth of The Mount School, York;
Larissa, Hollie and Emma, also of The Mount School;
Rachel and Christiane, again of The Mount School;
Georgina of Davison High School, Worthing;
Rosie of Davison High School, Worthing;
James of Hethersett High School.