Question

A watch manufacturer claims that its watches gain or lose no more than 8 seconds in a year. How accurate are these watches, expressed as a percentage?

Short answer

These watches have a percentage accuracy of 99.9999746%.

Step-by-step solution

Step 1. Definition of absolute error

The absolute error refers to the difference in the observed value and the true value of a quantity.

Step 2. Given data and assumption

The maximum absolute error in measurement of 1 year, $\Delta t=\pm 8\mathrm{s}$.

Step 3. Writing the formula for percentage error

The formula for percentage error is written as

$\mathrm{percentage}\mathrm{error}=\frac{\mathrm{absolute}\mathrm{error}}{\mathrm{expected}\mathrm{value}}\times 100$.

Step 4. Calculating the expected value

After 1 year, the expected value of measurement will be

$\begin{array}{c}\mathrm{expected}\mathrm{value}=1\times 365\times 24\times 60\times 60\\ =3,15,36,000\mathrm{s}\end{array}$

Step 5. Calculating the percentage error

Substituting the values in the formula for percentage error,

$\begin{array}{c}\mathrm{percentage}\mathrm{error}=\frac{\Delta t}{3,15,36,000}\times 100\\ =\frac{\pm 8}{3,15,36,000\mathrm{s}}\times 100\\ =2.54\times {10}^{-5}\%\end{array}$

Step 6. Calculating the percentage accuracy

Percentage accuracy can be written in terms of percentage error as

Percentage accuracy = 100 – percentage error.

Substituting the values in the above equation,

Percentage accuracy = 99.9999746%.

Thus, the percentage accuracy is 99.9999746%.