3 addtional machines will be required.
I use the following theorem .
When one quantity increases(decreases) while the other increases(decreases), set ratios equal.
When one increases(decreases), while the other decreases(increases), set products equal.
In the case number of machines increase while number of days decreases, hence we will set the products equal.
Let x be the number of machines that will finish the work in 8 days.
6*12 = x*8
x = 9
hence 9 machines are required to finish the work in 8 days.
No of additonal machines = 9-6 = 3