A rectangular plate has a length of 21.3 cm and a width of 9.8 cm. Calculate the area of the plate.
A straightforward question, which you multiply and arrive at 208.74 cm2. Taking significant figures into account, should the final answer be stated as 210 cm2?
The calculation of errors is important.
With the limited information that is given one can make an estimate of the error in the area by assuming the last digit in a length is in error by $\pm 1$.
However this example shows that caution must be applied when using the idea that the number of significant figures in the product cannot be greater than the number of significant figures for each of the quantities being multiplied together.
In the mathematical sense $98$ is a length given to two significant figures but when computing errors it is better to use the fractional error which in this case is $1/98$.
Using fractional errors shows that, although $98$ and $102% are quoted to differing numbers of significant figures, in terms of accuracy they are almost the same.
A simple estimate of the error is found by just adding the fractional errors and converting that into an actual error, eg $\pm \left (\frac {1}{98}+\frac{1}{213}\right )\times 208.74 \approx \pm 3$.
However this can be considered to be an overestimate as it assumes that both lengths are simultaneously maximum values or minimum values.
A better estimate of the error is $\pm\sqrt{\left (\frac{1}{213}\right)^2+\left (\frac{1}{98}\right)^2}\times208.74 \approx \pm 2$
You will note that in both methods the third figure in the product has some significance and so the value for the product should be quoted to three significant figures, ie $209\pm3$ or $209\pm 2$ depending on which method is used.
In the real world, measuring things perfectly isn't always possible. Those super precise numbers (like 208.74 cm²) are great for the calculator, but when it comes to the actual size of the plate, saying 210 cm² is more on point.
We know the length is around 21.3 cm and the width is around 9.8 cm, so considering those little measurement wobbles, the area is closer to 210 square centimeters. Think of it like rounding to make things easier to understand.
The final result should retain as many significant figures as there are in the original number with the least significant figures.
The answer to this question is therefore, 210 cm2 as the final result cannot have more than 2 significant figures.
Found a similar question which I have linked here for reference.