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When dealing with measurements in science and engineering, understanding how errors propagate is crucial. Specifically, when you multiply two quantities, the maximum fractional error in the product can be determined by simply adding the fractional errors of the individual quantities. Let’s break this down step by step.

Understanding Fractional Error

Fractional error is a way to express the uncertainty in a measurement relative to the size of the measurement itself. It is calculated using the formula:

• Fractional Error = (Absolute Error) / (Measured Value)

For example, if you measure a length of 10 cm with an absolute error of 0.2 cm, the fractional error would be:

• Fractional Error = 0.2 cm / 10 cm = 0.02 or 2%

Multiplying Quantities and Error Propagation

Now, let’s consider two quantities, A and B, with their respective fractional errors. If we denote the fractional error in A as εA and in B as εB, the product of these two quantities is C = A × B. The key point here is how the errors combine when you multiply:

• Maximum Fractional Error in C = εA + εB

Why Does This Happen?

This relationship arises from the way uncertainties behave in multiplication. When you multiply two numbers, the relative uncertainty in the result is influenced by the uncertainties in both numbers. Here’s a simple analogy:

Imagine you’re baking a cake. If you’re uncertain about the amount of flour (let’s say you might have 10% too much or too little) and you’re also uncertain about the amount of sugar (with a similar 10% uncertainty), the total uncertainty in your cake recipe is not just about one ingredient. Instead, it’s the combination of both uncertainties that affects the final product.

Example Calculation

Let’s say you have:

• A = 5.0 ± 0.1 (which gives a fractional error of 0.1/5.0 = 0.02 or 2%)
• B = 3.0 ± 0.05 (which gives a fractional error of 0.05/3.0 ≈ 0.0167 or 1.67%)

Now, when you multiply A and B:

• C = A × B = 5.0 × 3.0 = 15.0

The maximum fractional error in C would be:

• εC = εA + εB = 0.02 + 0.0167 ≈ 0.0367 or 3.67%

Practical Implications

This principle is not just theoretical; it has practical implications in fields like physics, engineering, and chemistry, where precise measurements are essential. By understanding how errors combine, scientists and engineers can better assess the reliability of their results and make informed decisions based on their measurements.

In summary, when multiplying two quantities, the maximum fractional error in the product is indeed the sum of the fractional errors of the individual quantities. This understanding helps in accurately reporting results and maintaining the integrity of scientific work.