Why is a over b plus c over d equal ad plus bc divided by bd always valid?

Answer 1

Because it can be derived from a large number of more basic principles applied to numbers.

Since a/b and c/d are defined, then b and d are non-zero.

Then a/b = ad/bd

since multiplying the numerator and denominator of a fraction by a non-zero number leaves it unchanged.

Similarly, c/d = bc/bd

[These steps implicitly assume the commutative property of multiplication ie there is no difference between premultiplication and postmultiplication]

Another property of numbers: If x = p and y = q then x+y = p+q [equals added to equals result in equals].

So a/b + c+d = ad/bd + bc/bd

and finally, using the distributive property of multiplication [by 1/bd] over addition,

= (ad + bc)/bd

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