Proving the Equality of Fractions Using Multiplicative Properties

[koshn]

Homework Statement

Prove that a/b=c/d if and only if ad=bc

Homework Equations

Multiplicative inverse property: (a)(a^1) = 1
Commutativity: ab = ba
Associativity: (ab)c = a(bc)
Transitivity: If a = b and b = c then a = c

The Attempt at a Solution

a/b=c/d=ab^1= cd^1. Multiplying both sides by (b)(d) I get, (a)(b^1)(b)(d) = (c)(d^1)(d)(b). Using the multiplicative inverse property, I get that ad=cb. And this is where I'm stuck. I don't know if this was the point of the proof. I'm not sure if the point was to prove (a/b)=(c/d)=ad=ac or if there is some other way to go about this.

 

[fzero]

You've proven that ad=bc if a/b=c/d. Now you need to prove the other direction, namely start with ad=bc and show that a/b=c/d.

 

[koshn]

Ok that makes sense. Thanks.