How to combine uncertainties in different quantities

One way to answer the question How many strides does it take to walk the Appalachian Trail is to estimate two quantities:

the length of the trail, L
the length of one stride, d

and then divide:


                             L
  number of strides  N  =  -----
                             d

Now, each estimate will have some uncertainty; for example, one group may guess


        L  =  1200 km  +/- 100 km

        d  =  68 cm +/- 3 cm

 so 
        N  =  1,765,706  strides   (but how many digits are justified?)

How can we incorporate the uncertainties in each of our two quantities to figure out the uncertainty in the number of strides?

In this particular case (division), there is a relatively simple rule: add the fractional uncertainty in each value to find the fractional uncertainty in the result.


                           ΔL        100 km
  fractional uncertainty ------  =  -------   =  0.083
                           L        1200 km

                           Δd         3 cm
  fractional uncertainty  -----  =  -------   =  0.044
                           d         68 cm 


   so ...

                           ΔN     
  fractional uncertainty  -----  =  0.083 + 0.044  =  0.127
                           N

One can also think of this procedure in terms of percentages:

"The length of the trail is known to 8 percent, and the length of a stride to 4 percent, so the number of strides will be known to (8+4) = 12 percent."

Okay, now time for you to finish this example.

What is the proper uncertainty ΔN in the number of strides needed to walk the trail, given these estimates?
Write the number of strides N with the proper number of significant figures.

You can find a great deal more information on combining quantities by reading