1/A=1/B+1/C, solve for A

1/A = (1/B) + (1/C)

Multiply by A:

1 = A/B + A/C

Multiply by BC (which is the least common denominator):

BC = AC + AB

Factor out the A:

BC = A(C + B)

Divide by (B + C):

BC/(B + C) = A

1 / A = ( C + B ) / ( BC )

A = (BC) / ( C + B )

1/A = 1/B + 1/C = C/(BC) + B/(BC) = (C+B)/(BC)

so A = (BC)/(B+C)

A = (B C)/(B + C)