A certain number N divided by one more than itself yields one fifth.
Another number M divided by one more than itself yields one fifth of itself.
It is quite easy to find the product M*N.

Try to do it " by heart" i.e. no calc, no pen, no paper.
.

Not very sure whether this qualifies as a "by heart" solution, but here goes:

We have:

N/(N+1) = 1/5 ..... (i)
M/(M+1) = M/5 .....(ii)

From (ii), we have: M+1=5

Substituting this in both sides of (i):

(M*N + N)/(N+1) = (1/5)*5
-> M*N+N = N+1
-> M*N = 1 QED 

Edited on January 10, 2012, 4:01 am

Posted by K Sengupta on 2012-01-10 03:53:26