Removing a number from one group and adding it to another seems as if it must raise one average and lower the other. But there is an interval where exactly the opposite happens.
The Card Between Two Averages
Riddle statement
There are two boxes containing numbered cards.
The average of the numbers in box A is greater than the average of the numbers in box B.
Can there be a card in A such that moving it from A to B raises the average of both boxes?
Show solution
Solution
Yes.
Let $m_A$ be the average of A and $m_B$ the average of B, with $m_A>m_B$.
If A contains a card of value $x$ such that:
$
m_B<x<m_A,
$
then two things happen at once:
- removing from A a value below its average raises A's average;
- adding to B a value above its average also raises B's average.
For example:
- A contains 60 and 100, with average 80;
- B contains 40 and 50, with average 45.
If the card 60 is moved from A to B:
- A is left with average 100;
- B is left with average 50.
Both averages have increased.
Answer: yes. It is enough to move from A to B a card whose value lies between the two averages.