For there to be no cash flow effect, the total value of the shares after the rights issue must be equal to the total value before the rights issue.
So…if they had (say) 1000 shares before the rights issue, they would have been worth $4000. To be worth $4000 after the rights issue, then they must then own 4000 / 3.85 = 1039 shares.
So they must have taken up 39 of the 200 shares they were entitled to, which is 39/200 = 19.5%.
(You can choose any number of current shares – the answer will be the same)