We are using quick sort here.
To sort 1000 names no of comparsion=$1000log_{2}\left ( 1000 \right )$
To sort 100 names no of comparsion=$100log_{2}\left ( 100 \right )$
So $1000log_{2}\left ( 1000 \right )$ comparison can be done in 100 sec.
Let the minimum time to sort 100 names is $x sec$
$1000log_{2}\left ( 1000 \right )$=$100 sec$
$100log_{2}\left ( 100 \right )$=$x sec$
So $x $=$\frac{100log_{2}\left ( 100 \right )\times100 sec}{1000log_{2}\left ( 1000 \right )}$
So $x $=$6.67 sec$
Minimum time needed to sort 100 names is 6.67 sec.