Verified answer

σz/z = σx/x + σy/y is an appromximation sometimes used in elementary work; it provides an upper limit on the value of σz/z.

But a more accurate formula is:

(σz/z)² = (σx/x)² + (σy/y)²

and I guess that's what you are meant to use.

I think it's because just adding the two fractional uncertainties overstates the uncertainty in z.

The absolute uncertainty in z is

σz = √[ (dz/dx∙σx)² + (dz/dy∙σy)² ]

which is where I always start thinking about questions like this. Note that the derivatives are actually partial derivatives.

I don't know how much calculus you've had (if any), so I'll point out that

dz/dx = y

dz/dy = x

so

σz = √[ (y∙σx)² + (x∙σy)² ]

then divide both sides by z = xy, yielding

σz/z = √[ (σx/x)² + (σy/y)² ]

which isn't what your manual says, so I guess I'm on the side of whoever told you the manual was wrong.