Expert: Josh - 4/4/2007

Question
QUESTION: Hello:

In order to invert 4/3,  simply invert the fraction to 3/4.
However, 4/3 can be expressed as 1 1/3.

Is it possible to invert 1 1/3 to 3/4 without first changing 1 1/3 to 4/3?

For example, 1 1/3 can be expressed as 3/3 + 1/3. 3/3 inverted becomes 3/3, and 1/3 inverted becomes 3/1. 3/3 + 3/1 = 12/3 = 4. This does not become the inverted fraction 3/4.

I thank you for your reply.

ANSWER: Hello,

The answer is no. It doesn't work like that.

Regards

---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for your reply.

Is there a reason why "It doesn't work like that."?

Answer
This is because the inverse of the sum of two fractions is not the same as the sum of the inverse of the two fractions.

Consider A/C + B/C. This may be written as a single fraction as (A+B)/C. Its inverse is C/(A+B).
This, C/(A+B), is not the same as (C/A) + (C/B). The latter is equivalent to (BC+AC)/AB.

The condition under which the former (inverse of the sum of two fractions with a common denominator) is equivalent to the latter (the sum of the inverse of two fractions with a common denominator) is given by
C/(A+B)=(BC+AC)/AB
ABC=(A+B)*(BC+AC)
ABC=ABC+(A^2)C+(B^2)C+ABC
-ABC=(A^2)C+(B^2)C
-AB=A*A+B*B
A^2+BA+B^2=0
A=[-B+sqrt(B^2-4B^2)]/2, or [-B-sqrt(B^2-4B^2)]/2

This problem has no real solution (unless we work with imaginary numbers), because the square root part is negative for all real values of B.

Conclusion: In general, the inverse of the sum of two fractions is not the same as the sum of the inverse of the two fractions.