#! perl
use strict;
use warnings;

my $start_day = '1/17';

while (my $payperiod = <DATA>)
{
    if ($payperiod =~ /$start_day/ .. 0)
    {
        print $., ' '. $payperiod;
    }
}

__DATA__
start: 1/3
aaa
end:   1/16
start: 1/17
bbb
end:   1/30
start: 1/31
ccc
end:   2/13