A metal sphere has a mass of 197 grams, correct to the nearest gram.

1. Find the lower bound of the mass rounded to the nearest gram

A mass of $197$ g correct to the nearest gram has lower bound
$197-0.5=196.5.$

1. Find the bounds needed for the greatest possible density-like value

Mass $=197\text{ g}$ correct to the nearest gram, so the greatest possible mass is
$197.5\text{ g}.$
Volume $=25\text{ cm}^3$ correct to the nearest cubic centimetre, so the least possible volume is
$24.5\text{ cm}^3.$

2. Calculate the greatest possible mass of 1 cubic centimetre

Greatest possible mass of $1\text{ cm}^3$
$=\frac{197.5}{24.5}$
$=8.061224\ldots$
$\approx 8.06\text{ g}.$