Error Calculation for Division $P/Q$

This problem involves calculating the combined random error for a quantity derived from the division of two other quantities, $P$ and $Q$. We are given the individual percentage random errors for $P$ and $Q$. The standard method for combining independent random errors uses the square root of the sum of squares of the individual fractional errors.

Calculating Fractional Errors

First, convert the given percentage errors into fractional errors:

• Fractional error in $P$: $f_P = \frac{\Delta P}{|P|} = 10\% = 0.10$
• Fractional error in $Q$: $f_Q = \frac{\Delta Q}{|Q|} = 2\% = 0.02$

Error Propagation Formula

For a quantity $Z = P/Q$, the fractional error $\Delta Z / |Z|$ is calculated using the formula:

$ \frac{\Delta Z}{|Z|} = \sqrt{ \left( \frac{\Delta P}{|P|} \right)^2 + \left( \frac{\Delta Q}{|Q|} \right)^2 }$

Substituting the fractional errors:

$ \frac{\Delta (P/Q)}{|P/Q|} = \sqrt{ (f_P)^2 + (f_Q)^2 }$

$ \frac{\Delta (P/Q)}{|P/Q|} = \sqrt{ (0.10)^2 + (0.02)^2 }$

Determining the Resulting Error

Calculate the value under the square root:

$ (0.10)^2 = 0.01 $

$ (0.02)^2 = 0.0004 $

$ (0.10)^2 + (0.02)^2 = 0.01 + 0.0004 = 0.0104 $

Now, take the square root:

$ \frac{\Delta (P/Q)}{|P/Q|} = \sqrt{0.0104} \approx 0.10198 $

Converting to Percentage Error

To find the percentage random error in $P/Q$, multiply the fractional error by 100%:

$ \text{Percentage Error} = 0.10198 \times 100\% \approx 10.198\% $

Rounding to one decimal place, the percentage random error is approximately 10.2%.