Adult hat size

I’m in the process of knitting a hat from Noro Designer Mini Knits - Design 2 and I just cast on 113 sts and have knitted 6 rows of k1p1 and I can see that it will be too big. I know the gauge is correct am using size 6 for rib then 7 for the crown. How do you figure out how many sts should I cast on since the book uses 113sts?
I also have the same problem with kids mittens - they are too small and followed their instructions. Could it be the wrong type of yarn?
Thank you

Your pattern may say that you should use size 6 and 7 needles, but that doesn’t mean the gauge will be correct for how [I]you[/I] knit, and you’re right that if you use a larger or smaller yarn than in the pattern you’ll get a different size. There should be a gauge on your pattern, so you should knit a sample to see if you match the sts/inch. If there’s too few/inch then use a smaller needle, if there’s too many, use a larger needle. And the yarn does have to be the same type to come out the right size.