This question is very long but I really need help, precisous metals and gems are measured in troy weights in?

English System:
24 grains = 1 pennyweight (exact)
20 pennyweight = 1 troy ounce (exact)
12 troy ounces = 1 troy pound (exact)
1 grain = .0648 gram
1 carat = .200 gram

a.

1 troy pound = 12 troy ounces=240 pennyweight= 5760 grains.

1 grain = 0.0648 gram

5760 * 0.0648 = 373.248 grams.

1000 gram = 1 kilogram

373.248 / 1000 = 0.373248 kg

b. 1 troy ounce = 20 penny weight = 480 grains

1 grain = 0.0648 grams

480 * 0.0648 = 31.104 grams

1 carat= .200 gram

31.104 / .200 = 155.52 carats

c. 31.104 grams / 19.3 g/cm^3 = 1.61 cm^3

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