Calculate the value of the error (only) with two decimal places for: $(5.1 \pm 0.4) / (2.5 \pm 0.2)$

Question

Calculate the value of the error (only) with two decimal places for:

$Z = \frac {5.1 \pm 0.4} {2.5 \pm 0.2}$.

I tried finding $Z_{max} = \frac {5.1(1+0.078)}{2.5(1-0.08)}$ which gives $2.04 + 0.335$ and I thought $0.335$ would be the error, but I am wrong.

Please help to see where I got off the track. Thanks!

Answer 1

I get $Z_{max}=5.5/2.3\approx 2.3913$, while $Z_{nom}=5.1/2.5= 2.04$. Your answer has more than two places.

Answer 2

Notice that the numerator and denominator are both positive. This guarantees that $$Z_{\text{max}}=\frac{\text{numerator}_{\text{max}}}{\text{denominator}_{\text{min}}}$$ and $$Z_{\text{min}}=\frac{\text{numerator}_{\text{min}}}{\text{denominator}_{\text{max}}}$$

Therefore,

$$\frac{4.7}{2.7} \leq Z\leq\frac{5.5}{2.3}$$ $$1.74(07)\leq Z \leq 2.39(13)$$

We can also calculate $Z_\text{nom}=\frac{5.1}{2.5}=2.04$. Therefore, the error is:

$$\max\{|Z_\text{nom}-1.74(07)| , |Z_\text{nom}-2.39(13)| \}$$ $$=\max\{|2.04-1.74(07)| , |2.04-2.39(13)| \}$$ $$=\max\{0.29(93) , 0.35(13) \}$$ $$= 0.35(13)$$

With two decimal places, the error appears to be $0.35$.