A manufacturer claimed that at least 20% of the public preferred her product. A sample of 100 persons is taken to check her claim. With how small would the sample percentage need to be before the claim could legitimately be refuted? (notice that this would invovle a one-tailed test of the hypothesis)

The book mentions that the answer is but im not sure how they calculated this.

I tried using

but got really stuck and couldn't calculate it out, so not sure if im on the right track...