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Convert 200 mg to g.

$"1000 mg=1 g"$

$200cancel"mg"xx(1"g")/(1000cancel"mg")="0.2 g"$

Convert carats to grams to moles.

$"1.00 ct=0.2 g"$
$1"mol C=12.0107 g C"$

$1.00cancel"ct C"xx(0.2cancel"g C")/(1cancel"ct C")xx(1"mol C")/(12.0107cancel"g C")="0.02 mol C"$

Convert moles C to atoms C.

$1 "mol C"=6.022xx10^23 "atoms C"$

$0.02cancel"mol C"xx(6.022xx10^23"atoms C")/(1cancel"mol C")=1xx10^22" atoms C"$ (rounded to one significant figure)

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