Reciprocal Sums
Problem Statement
Investigation/Exploration of the Problem
To solve this problem, we need to know which two positive integers added together is 11.There are many combinations.
10 + 1 = 11
9 + 2 = 11
8 + 3 =11
7 + 4 =11
6 + 5 = 11
A reciprocal as defined as two numbers whose product is 1. For example, 6 and 1/6 are reciprocals because 6*1/6 = 1 and m and 1/m are reciprocals because m*1/m = 1.
Using this definition, we can find the smallest possible sum of their reciprocals using a spreadsheet.
Extensions of the Problem
The sum of two positive integers is x. In terms of x, what is the smallest possible sum of their reciprocals?
We will use “the Arithmetic Mean & Geometric Mean Inequality” which states that
½ (a + b) ³ Ö(ab)
Since a + b = 11,
When and only when a = b =, and
That means ab has maximum , when a = b =
Therefore,
or
(i.e. the smallest possible sum of the reciprocals is 4/11)
Thinking this algebraic solution and the logic behind it if we graph
(yellow)
(turquoise)
(purple)
for p = 11 ( where p is the sum of the numbers)
you may observe that all those three graph intersect at a single point which is 4/11.
Author & Contact
Laura Thomas and Kursat Erbas
Memorial Middle School
Conyers, GA 30016