How does the precision of a calculated answer compare to the precision of the measurements used to obtain it?

This question refers to significant figures. The reason we follow significant figure rules when performing calculations in science is that a calculated answer (which is an extrapolation of a manipulation of measurements) can only be as precise as the least precise measurement recorded. This means if you record measurements to a certain precision, the calculated answer at the most can only match the recorded level of precision as to balance the level of confidence in accuracy before and after calculations.

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