Determining the number of significant figures in a measurement is usually as easy as shown in the gold examples. You simply count the number of digits in the measurement. Simple!
Sometimes, however, zeros in the number can cause difficulties.
Suppose you claim that the amount of gold as measured on the 1-place balance (in second example) was 1.6000 g. After all, adding zeros to the end of a number doesn't change its value, does it?
Recomputing the value of the gold, we get
Rounding this off to the nearest cent, we get $40.10. A small difference from the original answer, and definitely a bonus if you are selling the gold.
However, claiming that on the one-place balance, the gold weighs in at 1.6000 grams would be quite wrong. That would imply a degree of precision which the instrument simply does not have.
In effect, the one-place balance is saying that the weight of the gold is closer to 1.6 g than to any other number the balance could give. In other words, the balance is giving an answer as best it can, but there may be better answers.