Find an error of C in each of the following :

Question

1. C=A*B 2. C= A/B

Answer 1

This question isn't clear. I'm assuming that if there are errors in A and B (which we'll call ε_A and ε_B) then we get a corresponding error ε_C in C.

(1) So we can write C+ε_C=(A+ε_A)(B+ε_B)=AB+Aε_B+Bε_A+ε_Aε_B.

The last term can be ignored because for small errors it will be negligible.

On the RHS we can replace AB by C, and we obtain an expression for ε_C:

ε_C=Aε_B+Bε_A.

(2) We can write C+ε_C=(A+ε_A)/(B+ε_B).

1/(B+ε_B)=(B+ε_B)^-1=B^-1(1+ε_B/B)^-1=(1-ε_B/B)/B approx.

So:

C+ε_C=(A+ε_A)/(B+ε_B)=[(A+ε_A)/B](1-ε_B/B)=(A-Aε_B/B+ε_A-ε_Aε_B/B)/B. We can ignore ε_Aε_B/B as negligible:

C+ε_C=(A/B)+(ε_A-Aε_B/B)/B=C+(ε_A-Aε_B/B)/B, ε_C=(ε_A-Aε_B/B)/B or (Bε_A-Aε_B)/B². Note that errors are usually prefixed by ± so the error in C will also be ±.

If errors are given in fractions ε_A=Ap_A, ε_B=Bp_B. For percentage errors ε_A=Ap_A/100, ε_B=Bp_B/100, where p_A and p_B are fractions or percentages as appropriate.

I hope that this is in accordance with the intention of the question.