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.
rz/pqr = q/pqr + p/pqr
rz = q + p
r = (p+q)/z
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)
Thank you so much!!
To prove the given equation, we need to show that both sides of the equation are equal. Let's start by simplifying the expression on the right-hand side:
1/pr + 1/qr
To add these fractions, we need a common denominator. The common denominator in this case is prq. So, let's rewrite the expression with the common denominator:
(q + p)/prq
Now, let's simplify the expression on the left-hand side of the equation:
z/pq
Since we need to find an expression for r in terms of p, q, and z, let's begin by rewriting r in terms of p, q, and z:
r = (p + q)/z
Now, substitute this value of r in the expression z/pq:
z/pq = 1/(p + q)/z
To divide by a fraction, we invert the fraction and multiply:
z/pq = z/(p + q) * 1/z
Simplifying further:
z/pq = 1/z * z/(p + q)
The z in the numerators cancels out:
z/pq = 1/(p + q)
Now, comparing the left-hand side and the right-hand side of the equation, we have:
1/(p + q) = (q + p)/prq
By cross-multiplying:
1 * prq = (q + p) * (p + q)
prq = pq + p^2 + q^2 + pq
prq = 2pq + p^2 + q^2
Now, let's simplify the right-hand side:
prq = 2pq + p^2 + q^2
Rearranging the terms:
prq = (2pq + p^2 + q^2)
Since we have the same expression on both sides of the equation, we have proved that:
z/pq = 1/pr + 1/qr, where r = (p + q)/z