+128408 Egyptian Fractions have the form : 1 / positive integer
1/3 = 1/a + 1/b
Let z = 3
a,b must be > 3
So let a = z + m
And let b = z + n
So we have
1/ z = 1/ [z + m] + 1/[;z + n]
1/z = [ 2z + m + n] / [(z + m) (z + n)] cross-multiply
(z + m) (z + n) = z (2z + m + n ]
z^2 + mz + nz + mn = 2z^2 + mz + nz
z^2 + mn = 2z^2
z^2 = mn so
3*2 = mn
9 = mn
So the possibilites for m,n are
m n
1 9
3 3
So....the possible fractions are
1/ [3 + 1] + 1/ [ 3 + 9] = 1/4 + 1/12
1/[3 + 3] + 1/[3 + 3] = 1/6 + 1/6
Only the first is what we need
So
1/4 + 1/12
Here is another way:
1 - Start with 1
2 - Divide 1 by 4 = 1/4
3 - Take one of the remaining 1/4
4 - Subdivide that second 1/4 into 3 parts, or:
5 - (1/4) / 3 = (1/4) x (1/3) = 1 / 12
6 - Add 1/4 in (2) above to 1 / 12 in (5) above, or:
7 - 1 / 4 + 1 / 12 = 1 / 3 - which is what you want.
