The ideal gas law is# PV=nRT#. #P# is pressure, #V# is volume, #n# is number of moles, #R# is a constant, and #T# is temperature in #K#.
If the volume increases from #4L# to #16L#, that is like multiplying the left side of the equation by 4. To keep both sides of the equation equal, the right side must also be multiplied by 4. The only thing allowed to vary in this problem is #n#, the number of moles (#P# and #T# are kept constant, according to the problem). So, #n# must also be multiplied by 4.
The "new" #n# is #3(4) = 12# moles.
However, the answer is #"not 12"# (they are trying to trick you!). The question asked "how many moles must be injected into the container". There are already 3 moles , so an additional 9 moles must be injected to give you a sum of 12 moles .
The answer is therefore #"9 moles"#.
