Reciprocal Sums

Problem Statement
 

The sum of two positive integers is 11. What is the smallest possible sum of their reciprocals?
 

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.

  1. 10 + 1 = 11

  2. 9 + 2 = 11

  3. 8 + 3 =11

  4. 7 + 4 =11

  5. 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.

Click here for 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

lthomas@rockdale.k12.ga.us