Number Problem 11 Solution

This is a new problem as asked by a anonymous visitor which read: "a number added to five times its reciprocal is -6. find the number "

Ok non-algebraically you can think of what numbers added together will make 6(of course they will be negative to make negative 6) ---  1 and 5  and 4 and 2 , and 6 and 0...  You can't have  a reciprocal of 0 and a reciprocal of 6 : 1/6 when multiplied by 5 makes 5/6ths and you know those added together :  -6 + - 5/6ths dont make -6.   Same thing goes for 2 and 3  --> 2 + 5/2 or  3 + 5/3 dont make 6.   1 and 5 however do work.   The reciprocal of 1 is 1 so 1 + 5/1 =6  and so does 5 + 5/5 = 6 (Of course negative again).   So there are 2 answers -1 and -5...

Algebraically...   -x + (1/-x)5 = -6  =

-x -5/x = -6  (multiply by -x)

x^2 + 5 = 6x

x^2 + 6x + 5 = 0

(x + 1) (x + 5 )    = 0 

So the answers are x = -1 and -5