The mass of a box measured by a grocer's balance is $2.30 \ kg$. Two gold pieces of masses $20.15 \ g$ and $20.17 \ g$ are added to the box. What is
$(a)$ the total mass of the box,
$(b)$ the difference in the masses of the pieces to correct significant figures?

(N/A) Mass of the box $= 2.30 \ kg$.
Mass of gold piece $I = 20.15 \ g = 0.02015 \ kg$.
Mass of gold piece $II = 20.17 \ g = 0.02017 \ kg$.
$(a)$ Total mass $= 2.30 \ kg + 0.02015 \ kg + 0.02017 \ kg = 2.34032 \ kg$.
According to the rules of significant figures for addition,the final result should be reported to the same number of decimal places as the measurement with the fewest decimal places. Here,$2.30 \ kg$ has two decimal places. Thus,the total mass is $2.34 \ kg$.
$(b)$ Difference in masses $= 20.17 \ g - 20.15 \ g = 0.02 \ g$.
According to the rules of significant figures for subtraction,the result should be reported to the same number of decimal places as the measurement with the fewest decimal places. Both $20.17 \ g$ and $20.15 \ g$ have two decimal places,so the result is $0.02 \ g$.