If you watch the free lecture on rights issues then I go through an example very like this and explain the logic.
Whatever happens, the value of their new shareholding together with the change in cash will be the same as the total value of their old shareholding.
For there to be no change in the cash, it means the value of their new shareholding at the TERP must be the same as the value of their old shareholding at the current share price (of $4). If you make the two totals equal, you should then be able to calculate how many new shares were taken up and therefore what the % is of the shares they were entitled to.