# Thread: Find individual ages of the ten persons?

1. ## Find individual ages of the ten persons?

In a group of 10 persons, each person is asked to write the sum of ages of all the other 9 persons. If all the ten sums form a 9-elements set{82,83,84,85,87,89,90,91,92}, then find the individual ages of the ten persons.

Please explain the method explicitly while finding the solution.Thanks.

2. Originally Posted by fardeen_gen
In a group of 10 persons, each person is asked to write the sum of ages of all the other 9 persons. If all the ten sums form a 9-elements set{82,83,84,85,87,89,90,91,92}, then find the individual ages of the ten persons.
If all ten sums are added together, the result will be the sum of the ten ages multiplied by 9 (since each person's age will have been counted nine times).

In the 9-element set{82,83,84,85,87,89,90,91,92}, one item must obviously be duplicated. The sum of the nine elements is 783, which is already a multiple of 9. So the duplicated element must also be a multiple of 9. The only possibility for this element is that it is 90. Thus 9 times the sum of the ages is 783+90 = 873.

That should be quite enough of a hint to enable you to solve the problem.

Edit. The problem says that "each person is asked to write the sum of ages of all the other 9 persons." The solution gives the age of the youngest person as 5. How many five-year-olds do you know who could correctly add nine numbers ranging from 6 to 15 and then write down the result? That makes the problem seem unrealistic to me.

3. Can you please explain how one item must obviously be duplicated?

4. Originally Posted by fardeen_gen
Can you please explain how one item must obviously be duplicated?
You are told that all ten people were asked to write a number. But the set of numbers only contains nine elements. So one number must have occurred twice.