Question

Consider the following calculation:

x=(b−c) / a

The following experimental values and their errors are obtained:

a = 26.9 , Δa = 4.9

b = 46.1 , Δb = 2.6

c = 24 , Δc = 3.9

What is the absolute error in x (Δx)?

Note we are not asking for the value of x itself, only its error.

Answer 1

Relation	Equation	limit error
Subtraction	Z = x - y	$\small \Delta Z = \Delta x + \Delta y$
division	Z =x/y	$\small \Delta Z = \left | z \right | (\frac{\Delta x}{x} + \frac{\Delta y}{y})$

so, error in subtraction

$\small \Delta Z = \Delta b + \Delta c$ = 2.6 + 3.9 = 6.5

Z = b - c = 46.1 - 24 = 22.1

now,

$\small \Delta a =$ 3.9  

now, $\small (\frac{\Delta Z}{Z} + \frac{\Delta a}{a})$    = $\small (\frac{6.5}{22.1} + \frac{4.9}{26.9})$ = ( 0.294 + 0.182) = 0.476

now , X = $\small \frac{b - c}{a}$    = $\small \frac{46.1 - 24}{26.9}$ = $\small \frac{22.1}{26.9}$ =   0.8216

now, absolute error in the final calculation

$\small \left | \Delta X \right | = \left | X \right | (\frac{\Delta Z}{Z} + \frac{\Delta a}{a})$ = (0.8216 $\small \times$ 0.476 )=  0.391

hence absolute error (|$\small \Delta X$| ) = 0.391