How do you use the distributive property with fractions?

1 Answer
Oct 26, 2014

Distributive property of multiplication relative to addition is universal for all numbers - integers, rational, real, complex - and states that
#a*(b+c)=a*b+a*c#

In particular, if we deal with fractions, when each member of the above formula can be represented in the form #x/y# where both #x# and #y# are integers, the distributive law works in exactly the same way:
#m/n*(p/q+r/s)=m/n*p/q+m/n*r/s#
where #m,n,p,q,r,s# are integers and denominators of each fraction #n,q,s# are not zeros.

If we know the distributive law for integer numbers and understand that a rational number #x/y# is, by definition, a new number that, if multiplies by #y#, produces #x#, the above formula for fractions can be easily proved by transforming fractions on the left and on the right to a common denominator #n*q*s#:
#(m*(p*s+r*q))/(n*q*s)=(m*p*s+m*r*q)/(n*q*s)#.
In this form the distributive law for fractions is a simple consequence of the distributive law for integers.