# The confusion of fractional error calculation

I need to find the focal length of a lens by using equation 1/u + 1/v=1/f I have: u= 50+-3 mm v= 200+-5 mm I calculate the value of f as 40mm. Now i need to find the uncertainty in this value. I have two approaches, but only the second one is correct. I do not know what is wrong with the first one.

FIRST APPROACH : since f=(uv)/(u+v) Delta f/f= Fractional error of f= fractional error of u+ fractional error of v + fractional error of (u+v)

From this the uncertainty is 4.7 mm

SECOND APPROACH:we have Fractional error of 1/f = fractional error of f So delta( 1/f) = delta(f)/f^2 (*)

Similarly (*) is true for u and v in place of f

We have : delta(1/f) = delta(1/u) + delta(1/v)

So delta(f)/f^2= delta(u)/u^2 + delta(v)/v^2

From this delta(f) is 2.1mm which is correct

What is wrong with my first attempt?

The problem with your first approach is that you are assuming that the uncertainties in $u$, $v$ and $u+v$ are independent, when clearly they are not, they are highly positively correlated (when they are all positive). Hence you overestimate the uncertainty.
I should just add that I think both of your approaches are incorrect if you understand the error bar to mean the standard deviation of your estimate. Independent uncertainties should be combined in quadrature. I get $\delta F= 1.9$ mm.
• Because $u+v$ depends on the values of $u$ and $v$ !? In your second example, presumably $g$ and $l$ are independent variables. – Rob Jeffries Jan 2 '16 at 12:18