Calculating Uncertainty in Scientific Measurements

[qorizon]

Homework Statement

[/B]
a=12.26 +/- 0.08
b=0.25 +/- 0.05
e=1.2 +/- 0.2

Evaluate the uncertainty of following calculations

Part 1. z= 5(ab/e)

Part 2. p= 3e - b

The Attempt at a Solution

I attempted part 1 like this
Δz/z = 5 (Δa/a + Δb/b + Δe/e)
...
Calculated Δz to be 20 but the answer is
Δz/z = (Δa/a + Δb/b + Δe/e)
...
Δz = +/- 5 (to 1 sf)

Then I saw part 2's answer like this:
Δp = 3Δe + Δb
...
Δp= +/- 0.7 (to 1 sf)

Now I'm confused. Why is it that in part 1 the constant 5 in not taken into account when calculating uncertainty, but in part 2 the constant 3 is included? Am I doing it wrong?

 

[Samy_A]

In part 2 you are computing an absolute uncertainty, so the absolute uncertainty on 3e is 3 times the absolute uncertainty on e.
In part 1 you are computing a relative uncertainty, so a constant doesn't matter.

Imagine (for part 1) that 5 is a measured value d, with d=5 +/- 0.
Then for the relative uncertainty of z we get:
Δz/z =(Δd/d + Δa/a + Δb/b + Δe/e)=(0+Δa/a + Δb/b + Δe/e) =(Δa/a + Δb/b + Δe/e)
A value with 0 uncertainty has no impact on the relative uncertainty, that's why you ignore multiplication by a constant when computing a relative uncertainty.

(http://web.uvic.ca/~jalexndr/192UncertRules.pdf)

 

[qorizon]

In part 2 you are computing an absolute uncertainty, so the absolute uncertainty on 3e is 3 times the absolute uncertainty on e.
In part 1 you are computing a relative uncertainty, so a constant doesn't matter.

Imagine (for part 1) that 5 is a measured value d, with d=5 +/- 0.
Then for the relative uncertainty of z we get:
Δz/z =(Δd/d + Δa/a + Δb/b + Δe/e)=(0+Δa/a + Δb/b + Δe/e) =(Δa/a + Δb/b + Δe/e)
A value with 0 uncertainty has no impact on the relative uncertainty, that's why you ignore multiplication by a constant when computing a relative uncertainty.

(http://web.uvic.ca/~jalexndr/192UncertRules.pdf)

Thank you for the really helpful explanation!