Question

Show that z/pq = 1/pr + 1/qr, where r = (p + q)/z

Finding decmpositions of a fraction into 2 units as indicated on a paprus written in Greek.

Answer 1

rz/pqr = q/pqr + p/pqr
rz = q + p
r = (p+q)/z

Answer 2

RS = 1/(pr) + 1/(qr) , if r = (p+q)/z
= 1/( p(p+q)/z) + 1/( q(p+q)/z)
= z/(p(p+q)) + z/(q(p+q))
= (zq + zp)/( pq(p+q))
= z(p+q)/(pq(p+q))
= z/(p+q)
= LS

This allows you to express any fraction , which has a factorable denominator , into two fraction each with a numerator of 1

e.g.
suppose we have 5/12

5/12 = 5/(3x4)
so using the above formula
z = 5
p = 3
q=4
then r = 7/5

so 5/12 = 1/(3(7/5)) + 1/4(7/5))
= 1/(21/5) + 1/(28/5)
= 5/21 + 5/28
= 5( 1/21 + 1/28)

Answer 3

Thank you so much!!