December 2015
Problem of the Month

A Number and its Reciprocal

A real number is one more than its reciprocal.   Find the number.

You must show your algebra to get credit and give exact answers (not calculator approximations).

You may use sqrt() for "square root of" and you may use ^ for "exponent" and you may use +/- for the "plus or minus sign."

Solution to the Problem:

There are two possible answers:
(1 + sqrt(5)) / 2 and (1 - sqrt(5)) / 2.

The initial equation is x = 1/x + 1

You then rearrange the equation into a quadratic by

1. Multiply both sides of the equation by x to get x^2 = 1 + x

2. Subtract x and 1 from both sides to get x^2 - x - 1 = 0

3. Using the quadratic formula
(where a = 1, b = -1, c = -1), you find that possible values for x are (1 + sqrt(5)) / 2 and (1 - sqrt(5)) / 2

These two solutions are approximately equal to: 1.6180339887498948482045868343656... and
-.6180339887498948482045868343656...

Several problem solvers mentioned that this is the Golden Ratio.   Keith Mealy sent in the link to Vi Hart's video about the Golden Ratio.
If you can stand her put downs of math teachers, Click here for the video

Correctly solved by:

1. James Alarie	Flint, Michigan
2. Alex Hornett	McCulloch Junior High School, Marion, Indiana
3. Keith Mealy	Cincinnati, Ohio
4. Chase L. Smith	Mountain View High School, Mountain View, Wyoming
5. Gage Covington	Mountain View High School, Mountain View, Wyoming
6. Sreeroopa Sankararaman	Singapore, Singapore
7. Madison Bindl	Mountain View High School, Mountain View, Wyoming
8. Cecilia Marino	Mountain View High School, Mountain View, Wyoming
9. Raquel Mosteiro	Mountain View High School, Mountain View, Wyoming

Send any comments or questions to: David Pleacher