A reasoning that justifies the following equation?

Question

In this link (it's just a picture), is an equation;

http://img.photobucket.com/albums/v638/ravenatic20/untitled1.jpg

Can you give a reason that justifies the equation, using the 6 general laws of arithmetic (assoc., dist., comm.) and the basic laws of fractions?

Answer 1

a/b + c/d = (ad + bc)/(bd)

Use definition of division:
==================
The number N = a/b is such that Nb = a
The number M = c/d is such that Md = c
The number K = (ad + bc)/(bd) is such that K(bd) = ad+bc

We must prove that N + M = K.

Now
(N+M)(bd) = N(bd) + M(bd) ... distributive law
... = N(bd) + M(db) ... commutative law for *
... = (Nb)d + (Md)b ... associative law for *, twice
... = ad + cb ... definition
... = ad + bc ... commutative law for *
... = K(bd) ... definition

Therefore,
K(bd) = (N+M)(bd), so K = N+M (unless bd = 0).

Answer 2

in case you divide a million by skill of three, you get a million/3. Now in case you multiply a million by skill of a million/3, you get a million/3. So branch is basically multiplication with the reciprocal. So branch by skill of c/d is comparable as multiplying by skill of d/c. So (a/b)/(c/d) = (a/b) * (d/c)

Answer 3

(a/b) +(c/d) =

(a/b)*(d/d) +(c/d)*(b/b) =

(a*d) / (b*d) + (c*b) / (d*b) =

(a*d) / (b*d) + (b*c) / (b*d) =

(a*d + b*c) / (b*d)