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Relative error is a measure of the uncertainty of a measurement in relation to the size of the measured value itself. It provides an indication of how large the error is when compared to the actual number you are trying to measure. The relative error formula is simple: \[ \text{Relative Error} = \left( \frac{\text{Tolerance}}{\text{Measured Value}} \right) \times 100 \% \] This equation tells us the fraction of error in terms of percentage with respect to the measured value. In our exercise, the goal was to ensure that the relative error was less than 0.1%. Using the relative error formula helps in evaluating precision. For scientific instruments, like an analytical balance, a small relative error indicates high precision in measurements.

Tolerance is the maximum permissible error in the measurement process, which means it is the extent to which a measurement may vary without affecting its validity. It symbolizes how much the actual value could deviate from the ideal. Analytical balances often have a specified tolerance level, indicating their limits. In this exercise, the tolerance was given as \( \pm 0.1 \text{ mg} \). A smaller tolerance value means that the instrument is capable of higher precision in its readings, minimizing uncertainty. When you have tight tolerance levels combined with a calculated relative error, it can greatly impact how precise the measurement is perceived. Thus, a balance with a small tolerance contributes to higher accuracy in mass measurement tasks.

Mass measurement is a primary application of an analytical balance in settings such as laboratories and scientific research. It involves determining the weight of an object to a high degree of precision. For accurate mass measurement, several factors are considered, such as:

• Instrument Calibration: Ensuring the balance is properly calibrated ensures measurements are accurate.
• Environmental Factors: Conditions like humidity, air currents, and temperature can affect measurements.
• Sample Handling: Proper preparation and placement of the sample avoid additional discrepancies.

In our exercise, mass measurement required finding the minimum mass (100 mg) under the constraints of specific relative error and tolerance to ensure accuracy. This highlights the importance of understanding the capabilities and limitations of measurement instruments.