Adding unit fractions +

 New Android app:  Adding Unit Fractions +

4/5 

The app proposes 21 challenges to overcome.
Obtaining the proper fractions listed at the top of the application, 

adding two or three unit fractions.

Each proposed proper faction has a variable number of solutions.
And different levels of difficulty
You can not repeat unit fractions with the same value.
In the app you'll find a button to delete all the solutions found in the current problem, and to start from scratch.
The littlest unit fraction used in this app is 1/28.
The program is designed to show the usefulness of the subtraction of fractions in solving such problems.

Some hints: 

In the Rhindt Mathematical Papyrus (RMP) in 1650 BC the scribe Ahmes
copied the now-lost test from the reign of the king Amenemamhat III .

The first part of the papyrus is taken up by the 2/n table.

The fractions 2/n for odd n ranging from 3 to 101 are expressed as sums of unit fractions. 

In this app you can build some of the Ahmes  decompositions ( 2/3 , 2/5,2/7, 2/9 ) and 

the discarded ones by him also.

The app allow to decompose also: 

3/4, 3/5, 4/5, 5/6, 3/7, 4/7, 5/7, 6/7, 3/8, 5/8, 7/8, 4/9, 5/9, 7/9, 8/9, 3/10, 7/10, 9/10.

You can use the knowledge acquired solving the 2/X decompositions to solve the rest of the problems

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At first glance we can try the most elementary mechanisms: Subtraction of an essay:

2/3:

2/3 - 1/2 = 4/6 – 3/6 = 1/6; 2/3 = 1/2 + 1/6.

2/3 - 1/4 = 8/12 - 3/12 = 5/12; 2/3 = 1/4 + 5/12 = 1/4 + 4/12 + 1/12 = 1/4 + 1/3 + 1/12.

2/5 – 1/3 = 6/15 – 5/15 = 1/15; 2/5 = 1/3 + 1/15.

2/5 – 1/4 = 8/20 – 5/20 = 3/20; 2/5 = 1/4 + 3/20 = 1/4 + 2/20 + 1/20 = 1/4 + 1/10 + 1/20.

2/7:

2/7 – 1/4 = 8/28 – 7/28 = 1/28; 2/7 = 1/4 + 1/28.

2/7 – 1/6 = 12/42 – 7/42 = 5/42; 2/7 = 1/6 + 5/42 = 1/6 + 3/42 + 2/42 = 1/6 + 1/14 + 1/21

2/9:

2/9 – 1/6 = 4/18 – 3/18 = 1/18; 2/9 = 1/6 + 1/18.

To solve the basic problems of the 2/n table, Milo Gardner 

in the Wolfram's Math World suggests this basic rule first published in 2002:

2/(p*q) = (2/A)*(A/(p*q))  where A=(p+1)

Rule applied: 

2/(3*1) = (2/(3+1))((3+1)/(3*1))= (1/2)*((3/3)+(1/3))=(3/6)+(1/6) = 1/2+1/6.

2/(5*1) = (2/(5+1))((5+1)/(5*1))= (1/3)*((5/5)+(1/5))=(5/15)+(1/15) = 1/3+1/15.

2/(7*1) = (2/(7+1))((7+1)/(7*1))= (1/4)*((7/7)+(1/7))=(7/28)+(1/28) = 1/4+1/28.

2/(9*1) = (2/(9+1))((9+1)/(9*1))= (1/5)*((9/9)+(1/9))=(9/45)+(1/45) = 1/5+1/45

2/(3*3) = (2/(3+1))((3+1)/(3*3))= (1/2)*((3/9)+(1/9))=(3/18)+(1/18) = 1/6+1/18

From the basic 2/n table, we can afford easily the solution of the other problems:

3/4 = 2/4 + 1/4 = 1/2 + 1/6 + 1/4

3/5 = 2/5 + 1/5 = 1/3 + 1/15 + 1/5.

4/5 = 2/5 + 2/5 = 2/3 + 2/15 = 1/2 + 1/6 + 2/15 = 1/2 + 5/30 + 4/30 = 1/2 + 6/30 + 3/30 = 1/2 + 1/5 + 1/10.

5/6 = 4/6 + 1/6 = 2/3 + 1/6 = 1/2 + 1/6 + 1/6 = 1/2 + 1/3.

3/7 = 2/7 + 1/7 = 1/4 + 1/28 + 1/7.

4/7 = 2/7 + 2/7 = 2/4 + 2/28 = 1/2 + 1/14.

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