How Is Uncertainty Calculated for (a+b)/(c+d) When a=b=c=d?

[lavster]

Homework Statement

what is the uncertainty of $$\frac{a+b}{c+d}$$ if a=b=c=d and $$σ_a = σ_b =σ_c=_d$$

Homework Equations

$$σ_{a+b}=√(σ_a^2+σ_b^2)$$

$$σ_{\frac{a}{b}}=√((\frac{σ_a}{a})^2+(\frac{σ_b}{b}^2))$$

The Attempt at a Solution

since a=b=c=d

$$σ_{a+b}=√2 σ_a$$

$$σ_{\frac{a}{b}}=√2 \frac{σ_a}{a}$$

so $$σ_{\frac{a+b}{c+d}} = \frac{√2 √2 σ _a}{2a} = \frac{σ_a}{a}$$

is this correct?!

Thanks

 

[gneill]

The result looks good to me!

 

[lavster]

Great thanks :) any idea why it is the same as the uncertainty in a single measurement... Just doesn't seem right to me! Xxx

 

[BvU]

It is NOT the same as the uncertainty in a single measurement! That would be $\sigma_a$. Since you are evaluating a ratio, only relative errors matter. The factors $\sqrt 2$ and 2 just happen to cancel.

You can repeat the exercise with ${a+b+c}\over e+f+g$ and see what happens...